The Lies of Richard Dawkins, Episode 6: Saint Thomas Aquinas

Great Debates
441

I was sure this was a simple mistake. That you would notice that you had assumed a number system and the rules of operators, and admit it, (like I recognized I had assumed we were talking about sound systems,) and we could just laugh about it. But you really don’t know how to form a field, do you? You don’t know that you don’t need numbers? (The 0 and 1 in the axioms are not defined as numbers, but as additive and multiplicitive identities.)

[QUOTE=Indistinguishable]
I don’t understand what you mean by any of this. Here are the field axioms:
There are operators (+) and (*) and numbers 0 and 1 [and other numbers formable from those; e.g., 2 as 1 + 1, and so on].
[/QUOTE]

You’ve added the statement I’ve made bold. It’s nowhere in the axioms, can’t be proved from them, and isn’t necessary to fulfill them.

[QUOTE=Indistinguishable]
For all numbers x, x + 0 = 0 + x = 1 * x = x * 1 = x.
For all numbers x and y, x + y = y + x and x * y = y * x.
For all numbers x, there exists a y such that x + y = 0.
For all numbers x, if x is not equal to 0, then there exists a y such that x * y = 1.
For all numbers x, y, and z, x * (y + z) = xy + xz.

These most definitely speak about the operators (+) and (•), and numbers such as 2. And why do you say the field axioms are in no way a formal system?
[/QUOTE]

A field is a formal system, but you don’t have a field until you choose a set and set the operation of the functions + and *. Check finite fields and field theoryfor some examples of fields without the usual concepts of numbers or addition and multiplication.

example:

  • | 0 1 × | 0 1
    –±— --±—
    0 | 0 1 0 | 0 0
    1 | 1 0 1 | 0 1 a field that satisfies all the axioms, yet, 1+1 = 0. No 2 in sight.

And if you check out F4, you may be inclined to say, “it’s just using other symbols, it’s still about numbers.” Ok, replace A with 2, and B with 3. This still makes it a perfectly reasonable field. But you’ll notice that 1+2 = 3, 1+3 = 2, and 3*3 =2.

  • | 0 1 A B × | 0 1 A B -> + | 0 1 2 3 × | 0 1 2 3
    –±------- --±------- --±------- --±-------
    0 | 0 1 A B 0 | 0 0 0 0 0 | 0 1 2 3 0 | 0 0 0 0
    1 | 1 0 B A 1 | 0 1 A B 1 | 1 0 3 2 1 | 0 1 2 3
    A | A B 0 1 A | 0 A B 1 2 | 2 3 0 1 A | 0 2 3 1
    B | B A 1 0 B | 0 B 1 A 3 | 3 2 1 0 B | 0 3 1 2

[QUOTE=Indistinguishable]
I still don’t understand what you think makes the field axioms different from any other axiom system, but let me point out that whenever the axioms T neither prove nor disprove A, it is because there are two different possible models of those axioms, one where T and A both hold, and one where T and NOT A both hold. This is not special to the field axioms;
[/QUOTE]

But, you can’t ask questions about A in the system until you define it in the system. In F2, a field with 2 elements, or F4, (4 elements, but not numbers,) 2 doesn’t exist, and you can’t ask ‘is there an x such that x*x =2?’ It’s not undecidable, it’s inconceivable. And in every field I’ve seen where 2 exists, the question has a definitive answer. Either, ‘no, it can’t be done,’ or ‘yes, and the value or values can be found.’

And proposing a concatenated, bastard, system consisting of multiple fields doesn’t create undecidable results either; it creates inconsistent results. If you merge the rational and irrational fields, (and still keep their operations separate, since the irrationals include the rationals,) you get two definitive answers: ‘no, it can’t be done’ in the rationals, and ‘yes, it can, (and it’s about 1.4142,)’ in the rationals. If you add in my F4 system with elements 0,1,2,3, the answer is: ‘yes, and it’s 3.’ Perfectly decided individually, but inconsistent when considered together.

[QUOTE=Indistinguishable]
As a similar example, suppose I were to tell you “An infinitesimal is a number so small that, no matter how many times you add it to itself, the result is never bigger in magnitude than 100.” You might say “Whoa, that’d have to be a pretty crazy number. Can you actually prove that such things exist?” And I’d say “Yeah, sure. 0 is infinitesimal, for example.” Your response at this point would probably be “Oh, heh. I guess it’s not necessarily so crazy a notion after all, since it covers such trivialities. Actually, now that I see your proof, I realize that the established result is actually kinda boring; I was thinking of it as more mind-blowing than it actually is.”.
[/QUOTE]

An infinitesimal is usually defined as non-zero. But even if we excuse that, saying ‘the result is trivial,’ and then not bothering to think about infinitesimals would completely miss the massive, mind-blowing, implications of limits: the sum of infinite non-zero terms that never exceed a finite, (and possibly non-zero,) value. And that’s what you’re proposing with GIT, and (by the implication of your analogy,) with infinitesimals: dismissing the implications.

You’re saying the implications aren’t mind-blowing because the example is trivial. But GIT was a new result. I can ask questions about the number 2 for a system without a concept of 2, but I’m doing it outside the system. (except for questions like, ‘does 2 exist in this system?’. That is asking about a specific symbol for a system with a concept of symbols. and we expect an answer.) When we have a system with the concept, and we ask a question about that concept inside the system, mathematicians always expected there to be an answer, ‘yes, it’s possible,’ or ‘no, it’s not,’… until GIT.

442

You know, I almost didn’t respond to you in the first place for 3 basic reasons:

  1. The claim of ‘strawman,’ as typically used by all sides on this board, usually seems to mean, ‘I’m going to claim that this has been disproven without showing how or providing any argument myself.’ I roll my eyes almost instantly when I see it.
  2. The term ‘sky gods.’ What do you mean? Pigeons? They don’t seem very god-like, and I haven’t seen anything else in the sky that is either. So, apparently, I don’t believe in ‘sky gods’ either. Or, are you trying to replace one concept with a ridiculous one that you have an argument against, and claim the same argument holds for the broader concept?
  3. The almost complete lack of argument in your posts.

You said virtually the same thing a dozen times. It doesn’t matter how many times you repeat, “incidental,” it never becomes an argument.

[QUOTE=ch4rl3s]
then all those things we conceive/invent aren’t purposes either.
[/QUOTE]

You seem to agree that purpose can be created, but I can’t even argue with you about it because you don’t provide an argument. I have no clue how or when you believe purpose comes into being. You’re claiming everything else is incidental, why wouldn’t your conceived purpose be as well? Just the consequence of the firing of neurons in your brain that mean nothing.

When does purpose come into being?

Does your eye have a purpose? Does it collect light and transmit signals about the properties of that light to your brain? Does it exist in your body for that purpose? And you don’t need a brain for light sensing cells to have that purpose.

I’m saying that DNA and enzymes and proteins sitting on a rock waiting for the tide to wash them together and randomly interact doesn’t evince any purpose. Just like everything else mentioned: The Sun, Earth, rivers, etc. But once those proteins, enzymes and DNA came together inside a casing, and the DNA had the instructions to produce it all. It’s no longer random interactions, the interactions are nearly certain. The whole has become greater than the sum of its parts. It didn’t come together for any purpose, but it is now life. It has endowed itself with purpose.

[QUOTE=Diogenes the Cynic]
You’re allowed to invent a purpose fr your own life. Life as a whole does not exist for any discernable reason, though, nor is there any demonstrable reason why it needs to be perpetuated… It doesn’t matter if it keeps going. It doesn’t matter if it stops. The universe doesn’t care…
[/QUOTE]

And if you ‘invent’ a purpose for your own life, it still doesn’t matter if you accomplish it or not. The universe doesn’t care in that case either. So, are you claiming that that purpose isn’t purpose either? Even though you proposed it as a purpose?

Do you attempt to accomplish your purpose? Now that you have invented a purpose, does the universe suddenly care? No? Well then, why can’t life have a purpose, even if the universe doesn’t care about it. Why can’t it attempt to accomplish that purpose?

443

It does, the very first time, to anyone with half a wit to understand very simple concepts.

444

You’re simply equivocating between two distinct meanings of ‘purpose’ – one being more or less equal to ‘function’, and the other closer related to ‘meaning’. Life is purposeful in that it has a function it achieves, that of copying genetic information. Life is purposeless in the sense that it has a higher goal, i.e. some classically conceived ‘meaning of life’. Similarly, the eye has a function it fulfils, but no pre-determined goal-oriented purpose. It wasn’t created with the intent of serving as a sensory organ, and life wasn’t created with the intent of perpetuating itself. It’s just still here because that’s what it happened to do.

445

I had already admitted that it wasn’t scientific. So, what kind of evidence are we left with? If those are the two mentioned, I admitted it wasn’t scientific and actually used legal evidence as the example, which do you think I was talking about? Why do you assume I meant scientific when I had already denied that? Is that the only way you can argue an alternate point? To reject what I actually say?

Is legal evidence evidence from which a rational person can come to a conclusion?

And at some point, unless you perform the experiments yourself, (and who does that anymore?), we all take the word of others. We decide whose evidence is credible and come to conclusions. And forensic experts have been known to fabricate ‘evidence’ to further their own careers, or to ‘further justice.’ And other ‘scientists’ have been known to do that too. And sometimes, they just get it wrong. And yet, on the whole, *I find science and scientists in general, very credible.

And we’re all sometimes caught by that. I remember when the ‘cold fusion’ results first came out. And my physics professor discussing how this might be happening. And the conjectures that quantum tunneling would almost certainly be involved. And it was exciting to think it was possible. And I remember the (slight) disappointment when the results couldn’t be replicated.

*(and if people keep claiming I hate science, or something similar, this is the statement they will just ignore.)

446

I’ve never understood what’s so confusing about words like purpose or meaning. They are inherently subjective. There’s no talking about “the” purpose or “the” meaning of something. It’s ALWAYS the purpose SOME PARTICULAR BEING has for something or the meaning that SOME PARTICULAR BEING finds in something. If there exists a god that has a purpose for my life, that’s nice, but that’s ITS purpose, not THE purpose of my life. Nor does it’s purpose transitively become mine (though, of course, it could: I could find it’s purpose for my life very meaningful. But I wouldn’t necessarily find it so, especially if its purpose was “to be an anti-gay asshole to gay people”).

Your parents may see your purpose in life as to become a doctor. But you might find no purpose or meaning in that goal at all, and find art meaningful instead.

447

Here, finally, we have an argument. And a pretty good one. You might actually be able to convince me. Of course, (and about time,) this is only the start of the discussion.

Life not only has a function it achieves. It has a function it attempts to achieve. And from what I can tell of what Diogenes said, in the same way that any person can attempt to achieve a purpose. So, what does it matter if it’s not a ‘higher’ goal? And why does a goal-oriented purpose need to be predetermined? It’s the only natural purpose in the universe. The function of your eyes, ears, other organs are in service of it. They still exist because they helped life fulfill its purpose. Almost every purpose people attempt seem to actually be in service to it if you look closely. (Or should I say, ‘the purposes most people actually attempt,’?)

448

Those say two different things, to my mind; life can very well achieve the function of genetic information transfer only – even though you don’t seem to like that word – incidentally. It’s perfectly possible to conceive of a life form that doesn’t reproduce, though it probably won’t exist for very long. That’s essentially what I think Dio is saying – the reproduction thing isn’t necessary for life, life doesn’t exist with the express intent of fulfilling this ‘purpose’; it just happens to do.

Well, it would not be goal-oriented if it weren’t set up to achieve its goal; achieving it would be incidental to its existence. There’s a fundamental difference, to me, in whether or not things were set in motion with the express intent of ending up somewhere, or if they just ended up there by chance.

I’d say that purpose of the kind that entails meaning only comes into existence with cognition, because only then can something be designed to achieve some form of goal; such a thing requires a conception of the future, and the ability to plan for it. And that’s how you can give your life a purpose: set yourself a goal, and work towards achieving it. This is, however, not the same kind of ‘purpose’ life has in reproduction, since the latter lacks intentionality.

449

Um, stating your conclusion without supporting it does not an argument make.

[QUOTE=Half Man Half Wit]
life can very well achieve the function of genetic information transfer only – even though you don’t seem to like that word – incidentally.
[/QUOTE]

I have no problem with the word. I have a problem with someone using his conclusion as his entire argument. (over and over again.)

[QUOTE=Half Man Half Wit]
It’s perfectly possible to conceive of a life form that doesn’t reproduce, though it probably won’t exist for very long. That’s essentially what I think Dio is saying – the reproduction thing isn’t necessary for life.
[/QUOTE]

[QUOTE=ch4rl3s]
And I don’t claim life must exist for a purpose. I simply state that it does exist for this purpose. Although this purpose is inherent to the concept of life as we know it. It’s possible for life to exist without this purpose; for instance, not containing the DNA to reproduce. It won’t last long, and we don’t see examples of it.

[/QUOTE]

Yeah. Who said that?

I agree that DNA and proteins sitting on a rock waiting for the tide to wash them together constitutes incidental. But a cell sitting on the same rock doesn’t wait for anything to move the proper proteins over its DNA; it does it itself. And it doesn’t wait for the tide to move food over it. If there is no food on its bit of the rock, it moves to find food.

[QUOTE=Half Man Half Wit]
I’d say that purpose of the kind that entails meaning only comes into existence with cognition,
[/QUOTE]

It’s clear you mean “higher” cognitive functions here. But, I’m going to point out that the word can be used in another sense to indicate information processing. That which you think of as higher cognition is still just information processing. Life processes information in many ways. Some of them involve neurons.

If you are going to say that life striving to survive is purely incidental, then I can claim that your eye is purely incidental, (you’ve already said so, but I want to elaborate,) it came into being incidentally, and therefore, even though it now exists to process and transfer information, that processing is incidental. I can claim that your brain is incidental. It came into being incidentally, and any information processing it does is on the same incidental order as every other function of organisms. The neurons fire incidentally, other neurons respond incidentally, the thought that is produced is incidental. (I’m channelling Dio for a second here. But at least I’m producing a chain. One incidental leading to the next.)

[QUOTE=Half Man Half Wit]
There’s a fundamental difference, to me, in whether or not things were set in motion with the express intent of ending up somewhere, or if they just ended up there by chance.
[/QUOTE]

I see a difference there too. But then I see life itself as something different. Without that, I see a continuum of ever more complex behavior. The firing of your neurons is just a more complex form of information processing. (edit: information processing to decide on an action started with life, though.) It still came about incidentally whatever function it now performs. (That seemed to be the argument you put forth for everything I proposed to have a function. Why wouldn’t it apply to the brain, too?) The “thought” you have to put some money away for vacation is still just the effect of life “randomly” trying new things to acheive its purpose and really no different than a squirrel hiding nuts for the winter. They have exactly the same effect.

[QUOTE=Half Man Half Wit]
Life is purposeless in the sense that it has a higher goal, i.e. some classically conceived ‘meaning of life’.
[/QUOTE]

The sort of “meaning” you are proposing for cognitive thought doesn’t rise to that standard, either. You are proposing that each person picks some purpose, any purpose, (any purpose at all,) and the existence of that one, tiny, individual purpose choosen solely for that one, tiny, individual life somehow ranks higher than the existence of life itself. How does that happen, anyway?

450

I do know how to form a field.

Well, if we agree to call whatever is in the field “numbers”, then we’ve got numbers. Same with the axioms of Peano Arithmetic. But whatever. It doesn’t matter.

Sure it follows from the axioms; if you have 1 and you have +, then you have 1 + 1, which we can abbreviate as 2. Now, maybe, in some particular field, 2 = 0, or what have you, but I never said anything about that.

Anyway, I don’t care about fields as such. I just care about the axioms I gave. I gave you a particular set of axioms and noted that those axioms were incomplete, and that, yet, no one would be surprised or impressed by this fact. That’s all I meant to do with those axioms.

If we take “2” to mean “1 + 1”, we can ask the question no matter what, which is what I was doing. Anyway, sure, the question has an answer in every particular field, but you can’t deduce an answer just from the axioms I listed. Which was my point. It’s the exact same way with Goedel’s results: every model of Peano Arithmetic gives a particular answer to every question in the language of Peano Arithmetic; it’s just that the axioms themselves don’t give answers to every such question (for the precise reason that not all the models agree on all the answers).

I think you have been confused as to what I was doing by mentioning the field axioms. As I said above, I was never proposing taking separate fields and joining them together into some bastard system. I was saying “Look at these axioms; this very particular set of axioms I am listing. I know that these axioms are incomplete. How do I know these axioms are incomplete? I know this because these axioms are satisfied by many different models, and those models don’t all agree on all the answers to every question in the relevant language”.

Like I said, the field axioms (by which I mean a very specific set of axioms which I listed in my post) and Peano Arithmetic (by which I mean a very specific set of axioms which you could no doubt list as well) are exactly the same in this regard: every particular field and every particular model of Peano Arithmetic answers every question with some particular answer. However, the axioms themselves do not settle all the answers, because there are models which go one way and models which go another way. There are models of Peano Arithmetic in which Goodstein’s Theorem holds true and there are models of Peano Arithmetic in which Goodstein’s Theorem holds false, and so on, just the same as there are fields in which there exists a field element whose square is 1 + 1 and there are fields in which no such element exists. Peano Arithmetic’s incompleteness is not of some wholly new kind unexhibited by the field axioms; that’s not the interesting thing about Goedel’s result. The interesting thing is a simple criterion for immediately demonstrating that a wide class of axiom systems exhibit the same incompleteness. But it’s not a new phenomenon being witnessed in any of those cases; it’s just the same old thing as happens with the field axioms.

451

I have heard different definitions for universe. A different universe could be a bunch of stuff that doesn’t interact with anything in our universe, right?

452

ch4rl3s, since I seem to have somehow misled you when I tried to make this point the first time, forget I ever mentioned the word “field”, “numbers”, or what have you. Let’s just look at some particular axiom system:

[The language of discussion includes everything in single-sorted first-order logic with equality. It extends this with the constants 0 and 1, and binary operators (+) and ().]
The axiom system I want to look at contains the following axioms:
For all x, x + 0 = 0 + x = 1 * x = x * 1 = x.
For all x and y, x + y = y + x and x * y = y * x.
For all x, there exists a y such that x + y = 0.
For all x, if x is not equal to 0, then there exists a y such that x * y = 1.
For all x, y, and z, x * (y + z) = xy + x*z.
0 is not equal to 1
FIN

Whatever it can or can not be taken as talking about, that’s a perfectly cromulent system of axioms, right? I mean, I can demonstrate that some things follow from these axioms (e.g., that there exists an x such that x + x = x, that for all x, y, and z, (x + y) * z = x*z + z * y, that if 1 + 1 + 1 + 1 = 0 then 1 + 1 = 0, that 1 + 1 is not equal to 1 + 1 + 1, and so on).

But it turns out that there are some things which neither follow from these axioms, nor whose negations follow from these axioms (e.g., these axioms neither prove that there exists an x such that x * x = 1 + 1 nor that that there does not exist such an x). So what? Well, so that’s incompleteness, right there. We’ve fixed some language and and some axiom system within it, and we see that those axioms neither prove nor disprove everything within that language.

And if I were to start removing axioms and talking about an even more impoverished axiom system (say, let’s junk the third axiom above, and ask what we could prove from the rest), well, we’d be able to prove even less. There’d be even more statements which the axiom system could neither prove nor disprove. More incompleteness!

So incompleteness is not some rare transcendent phenomenon; it’s old hat. It happens all the time. It’s not, in itself, particularly shocking (unless you think that what was illustrated here is, in itself, particularly shocking?).

453

(Sorry, not that it matters, but I had intended to include in the example list of axioms also the axiom “For all x, y, and z, x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z”. Of course, this just means that what I gave is even more incomplete than what I meant to give…)

454

Sorry, I missed that.

You could make the same argument about fire, it seems to me. There’s no intentionality in the cell’s food finding, it’s a simple control loop.

If I get you right here, you’re essentially saying that, since everything ever since the first molecule learned the reproduction trick is just part of a mindless chain of cause and effect, then so is the decision to give your life a purpose. Well, I can’t disagree with that, but does this really invalidate that purpose? No matter whether or not your thought ‘I’m in my life gonna try to achieve X’ is just predicated by so many causes and effects, its effect in turn is that your life is now directed to accomplishing X. It’s the closest we – or anything – can get to setting a goal-oriented purpose (barring the existence of actual free will).

I’ve never really said anything about the ‘rank’ of purpose, I think. I don’t see a viable metric to compare the existence of life to my desire to be an astronaut (or whatever). I see life itself as purposeless, even though it accomplishes the function of reproduction; and I think that the individual can set a purpose for himself (or, say, some piece of machinery he constructs or similar) according to what I wrote above. I don’t see a contradiction.

Anyway, what this is really aimed at – or was, originally – is your assertion that Genghiz Khan was a ‘good atheist’ because he had a lot of offspring, since all the atheist life can have in the way of purpose is reproduction (never mind even the issue with the belief in god(s) or somesuch). If you’re not still claiming that, I guess I can live with the remaining differences between the two of us.

455

The part in bold is a DEFINITION and therefore a new axiom you’ve added.

[QUOTE=ch4rl3s]
You’ve added the statement I’ve made bold. It’s nowhere in the axioms,
[/QUOTE]

[QUOTE=Indistinguishable]
Sure it follows from the axioms;
[/QUOTE]

Previously you had essentially tried to define a successor function, S(x) to arrive at the numbers 2 to infinity. It was a DEFINITION and therefore a new axiom you had added. Now, I’ve done that. Most people who work with this stuff have. But, when it’s pointed out, most of us say, “no, I di… holy crap I did!” and we DON’T try to “fix” the situation by continuing to add more axioms.

[QUOTE=Indistinguishable]
if you have 1 and you have +, then you have 1 + 1, which we can abbreviate as 2.
[/QUOTE]

DEFINITION. And an axiom you’ve added.

[QUOTE=Indistinguishable]
If we take “2” to mean “1 + 1”,
[/QUOTE]

The same axiom, again.

[QUOTE=Indistinguishable]
…**and Peano Arithmetic **(by which I mean a very specific set of axioms which you could no doubt list as well)…
[/QUOTE]

So, you’ve looked at the Peano axioms? Did you notice they have several axioms to arrive at a successor function to define the natural numbers? The natural numbers that you started by ASSUMING existed in every field, (and still keep trying to add specific instances of,)

[QUOTE=Indistinguishable]
Anyway, I don’t care about fields as such. I just care about the axioms I gave.
[/QUOTE]

Might mean something if you had shown any knowledge of what an axiom was. Since you keep adding them willy-nilly with no idea you’re doing it.

[QUOTE=Indistinguishable]
I gave you a particular set of axioms and noted that those axioms were incomplete, and that, yet, no one would be surprised or impressed by this fact. That’s all I meant to do with those axioms.

[/QUOTE]

I think I know what you were trying to do. You also tried to compare what could be said about all fields, (which are questions from outside any formal system,) with what could be said within a field. And it was a ludicrous comparison.

Just like this was a ludicrous comparison:

To prove that Goedel’s proof isn’t profound because it’s trivial you posit a trivial example, but one that has severely profound implications!!! Do you realize how severe the implications of infinitesimals are? But I suppose we can ignore all that because your proof was trivial.

The interesting thing about Goedel’s result was that a sufficiently powerful system can’t be completed without becoming inconsistent.

BUT, a complete system doesn’t have to be powerful. There are complete systems that are more “impoverished,” (less powerful,) than ZFC or Peano. And that are nontheless, consistent. Minimal can be better for some purposes. Just because a system doesn’t have all the features of some more powerful system doesn’t mean it’s going to be incomplete.

456

I don’t presume to speak for Indistinguishable, but the way these are used in his argumentation makes them mere conventions, not axioms, that can either be adopted or not without influencing the point he’s making at all. Calling ‘1 + 1’ 2 in a shorthand way isn’t anything different from calling it ‘0’, or ‘elephant’, or ‘your mom’. It doesn’t say anything about the symbols that are used other than that they are used in such a shorthand representative fashion.

I think that’s actually the same thing that Indistinguishable is trying to demonstrate – many people think that the interesting result is that incomplete systems exist at all, but that’s not really anything surprising or deeply mysterious. It’s often represented as showing that ‘mathematics hasn’t got all the answers’, or some such pseudophilosophical nonsense, but that’s not the surprising result; that lies in the impossibility to find a concise set of axioms for all mathematics, or in the impossibility to carry out Hilbert’s program as it was originally conceived, or however you’d want to look at it.

457

Half Man Half Wit is correct about what I was doing with the notational shorthand of “2” for “1 + 1” and so on, but it doesn’t matter; I realized that part of the discussion was getting bogged down unnecessarily by details of my original presentation and so reformulated it into post #452, to which, I am happy to see, you (ch4rl3s) raised no objections.

Well, as I was getting at, how severe are the implications of the mere existence of infinitesimals when this is defined via “An infinitesimal is a number so small that, no matter how many times you add it to itself, the result is never bigger in magnitude than 100,” so that, though we may not realize it at first, even 0 counts as an infinitesimal? What once seemed like a profound statement may turn out rather mundane, once we see the trivialities it allows for. That was the point I was trying to make with this comparison.

Removing axioms from an axiom system will only cause it to be able to prove less. So, what I was saying was, if you start with an incomplete axiom system, and remove axioms (while keeping the language of statements one is dealing with the same), then the system one ends up with will necessarily be incomplete as well [it can’t prove anything the starting system didn’t].

You make oblique reference to the completeness of “impoverished” systems such as Presburger Arithmetic and so forth. But this doesn’t contradict what I was saying. Sure, you can think of Presburger Arithmetic as Peano Arithmetic with all the axioms about multiplication ripped out… and if you think about it that way, then Presburger Arithmetic is incomplete (in the sense that it can neither prove nor disprove certain statements; particularly, those axioms of Peano Arithmetic which dealt with multiplication). In order to think of Presburger Arithmetic as a complete system, we have to also narrow the scope of the language we are concerned with from that of Peano Arithmetic, so that we do not concern ourselves with statements which mention multiplication; in this case, Presburger Arithmetic would not just be Peano Arithmetic with some axioms removed, and all would be alright.

Anyway, it was just a very minor point, but hopefully you can see now what I was saying, which is pretty trivially obvious: removing axioms from an axiom system can at most result in being able to prove less, and thus in being even further incomplete. All the more reason why incompleteness abounds and is not particularly surprising in itself.

As Half Man Half Wit points out, this is already far beyond simply noting that some systems show some incompleteness (and thus why I objected to lumping Goedel’s results in as automatically related to, e.g., the incompleteness of ZFC with respect to the Continuum Hypothesis, which is really a very unrelated example of incompleteness, just as was the example of post #452).

But, sure, I basically agree with what you are saying here, except for one slight nitpick which is not really a vehement disagreement:
It doesn’t really have to do with how “powerful” the axiom system is; it has to do with the balance between how simple the axiom system is to express and how expressive the language of statements dealt with by the axiom system is. That is, the Goedelian incompleteness phenomenon will arise whenever the language dealt with by the axiom system is expressive enough to define the axiom system itself [in suitable fashion]. Thus, we could just as well say, the “weaker” (in the sense of being easier to express) the axiom system, the more vulnerable to incompleteness it is. But it would be best off not to say this either. It would be better to simply say “Given an axiom system for statements in some language, where the language is capable of expressing that axiom system itself, it cannot be both complete and consistent”. To get incompleteness, you want a “powerful” language (in the sense of being expressive) but a very “weak” actual axiom system (in the sense of being easily expressed).

To illustrate what I mean by this: the standard result we all know and love is that there is no computably enumerable axiom system which can interpret natural number addition and multiplication which is both complete and consistent. Great. But there’s two constraints here worth noting: not only that the language be “powerful” enough to interpret statements about <N, +, *>, but also that the axiom system be “weak”/“simple” enough that it can be enumerated by computer. Both of these constraints are necessary; otherwise, we could easily obtain consistent completeness (e.g., the axiom system consisting of all true statements about natural number addition and multiplication is certainly complete and consistent; it just happens to be too complicated to enumerate by computer). And, indeed, the two constraints match, because every computable function is definable (in a tedious way) using only statements about natural number addition and multiplication; so all that is really going on here is the observation that no axiom system can simultaneously be expressed in the language of <N, +, *> and correctly answer all questions about <N, +, *>.

Indeed, even Presburger Arithmetic, complete though it is, illustrates the same result: the axiom system of Presburger Arithmetic deals with the language of <N, +>, but cannot be expressed in the language of <N, +>. As always, no axiom system formulated in language L can correctly answer all questions about language L.

So the phenomenon is not really about the “powerfulness” of the axiom system; it’s a tug-and-pull involving both the “powerfulness” (expressiveness) of the language and the “weakness” (ease to express) of the axiom system. Whenever these match up, we have incompleteness. [Precisely because we could then ask “Does the axiom system being looked at answer ‘No’ to this question?” in the language being dealt with]

But, whatever. We don’t really disagree here. Where we’re going to disagree are on what the philosophical implications we should take from this are.

458

(Lest I be misinterpreted, by “Goedel’s results” here, I mean, of course, only Goedel’s Incompleteness Theorem. Other work by the same man (e.g., on the predicative cumulative hierarchy [“the constructible universe”, “L”]) is obviously directly relevant to the metamathematics of the continuum hypothesis, but, I would say, not significantly related to the incompleteness theorem)

459

Aaaaaaarrrrrgghhh!!!
No wonder I was getting so frustrated…
I see now what you were doing. You were changing definitions between posts, and even in the same post without telling me!

Completeness and incompleteness have mathematical definitions. Silly me to think that we would be using them for a mathematic discussion. In this post, you have used two separate definitions of complete…

[QUOTE=Indistinguishable]
…even Presburger Arithmetic, complete though it is…
[/QUOTE]

Mathematic definition.

A definition which “proves,” (since Presburger can be formed by removing axioms from Peano,) that Presburger is incomplete.

So, I’ve been thinking that you’ve been trying to say things like Presburger is both complete and incomplete. Imagine how frustrating that is? And it’s thoroughly bad form on your part.

Imagine how impossible it is to have a coherent conversation with someone who switches definitions without saying so. (This being the first time you’ve stated this definition, even though you’ve been using it extensively.)

I’ve just now realized you’ve also been switching between laymen and mathematic definitions of decidable and undecidable. Which is why I was trying to keep you talking about decidable within the language of the system. Because that’s how it’s defined mathematically. (The statement you are deciding has to be expressed in the language of the system.) And once again, truly bad form on your part.

It did matter. Because you started by claiming they followed from the axioms. That is not a convention to be accepted or rejected. ‘Following from the axioms’ means it exists in the system. And since it didn’t follow from the axioms, and I was assuming the mathematic definitions of complete and decidable, I could only assume you were forcing it into the system in order to ask questions about it which were “undecidable.”

Starting by claiming it is in the system and now claiming you’ve only been using it as a convention, (as well as being untrue,) is truly, truly, bad form. Maybe, this last time it can be termed a convention… if you ignore everything that came before. But I had no way of knowing that, since it couldn’t be termed a convention before, and you didn’t say.

[QUOTE]
I realized that part of the discussion was getting bogged down unnecessarily by details of my original presentation and so reformulated it into post #452, to which, I am happy to see, you (ch4rl3s) raised no objections.

I did object to one part. The one where you claimed that removing axioms makes things more incomplete. When I know that removing axioms can produce a complete, (math definition,) system. Since I had no way of knowing you were changing definitions. I didn’t see any error with the axioms, or those things that are true for all systems using them. I only objected to everything after that involving the term “incomplete.”

And it was very confusing and maddening to me when you showed obvious technical knowledge of math concepts then lapsed into what was such an obvious error, (using the definition of complete as I could only naturally assume we were using it.)

I now see that we really don’t have tremendously different views. But do you see how frustrating it is to be dealing with someone who doesn’t tell you until a dozen or so posts later that they’re using different definitions?

That’s only true if you start discussing math using the non-mathematic definition of complete. How could I have known you were doing this? You should have told me earlier that you were using a different definition. We could have saved weeks of frustration.
I still claim, however, that it’s two different things to talk about (non-math definition) undecidable from outside a system, and (math definition) undecidable inside a system. Outside a system, using a language not in the system, it’s trivial to produce something “undecidable.” What is the blah such that bloh-de-blih is bleh? But, (math definition,) undecidable, meaning using a statement that can be formed in the system is completely different.

For instance. Presburger doesn’t include multiplication, so it’s trivial to say it can’t answer questions about multiplication. But you’re asking the question outside the system. But, ZFC has a well defined notion of sets and which ones are possible within it. And not being able to determine if a certain set exists in ZFC, (like with the continuum hypothesis, and the informal question, “is there a set whose cardinality is between the integers and the reals?”,) would be like not being able to determine if the set of integers has a member between 10 and 11.

That’s why I objected to your analogy of just using the field axioms. You weren’t comparing like things. and we were using different definitions to compare. Which isn’t a valid comparison. It’s the laymans definition of “undecidable” to say “is there an x such that x*x = 2?” is undecidable from the field axioms. And the math definition of undecidable to say the continuum hypothesis is undecidable in ZFC. We’re not speaking the same language here.
It’s obviously an interesting question to determine what things are true for all systems that use certain axioms. But not being able to determine if something is true for all such systems isn’t the same as not being able to determine if something is true for one particular system. It’s not using the same language.

Sorry. Once again, running out of time to discuss anything else.

460

I don’t think I’ve been changing definitions. I’ve always been using “complete” to mean “either proves or disproves every statement [in the relevant language of statements]”. Using this definition, one can’t remove axioms from an incomplete axiom system and end up with a complete axiom system (unless one simultaneously thinks of oneself as narrowing the relevant language of statements, which is a separate issue from simply removing axioms).

I still don’t understand what you’re talking about with “It’s the layman’s definition of undecidable’ to say ‘Is there an x such that x*x = 2?’ is undecidable from the field axioms.” But remember that I want to chuck away the term “field axioms” since it caused so much confusion. Forget that I ever mentioned the word “field”. Ok?

I gave a list of axioms in post #452. I’ll give it again:

[The language of discussion includes everything in single-sorted first-order logic with equality. It extends this with the constants 0 and 1, and binary operators (+) and ().]
The axiom system I want to look at contains the following axioms:
For all x, x + 0 = 0 + x = 1 * x = x * 1 = x.
For all x and y, x + y = y + x and x * y = y * x.
For all x, there exists a y such that x + y = 0.
For all x, if x is not equal to 0, then there exists a y such that x * y = 1.
For all x, y, and z, x * (y + z) = xy + x*z.
For all x, y, and z, x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z
0 is not equal to 1
FIN

Alright, that’s a perfectly good axiom system, yeah?

Now, the mathematical definition of “incomplete” is “Either proves or disproves every statement [in the relevant language of statements]”. I claim that one statement in the relevant language is “There exists an x such that x * x = 1 + 1”, and that the above axiom system neither proves nor disproves this. Thus, it is incomplete, according to the mathematical definition of incomplete.

Do you object to this? If so, what is it that you object to?
A) The list of axioms above constitutes an axiom system, for the indicated language of statements
B) The mathematical definition of “incomplete” is “Either proves or disproves every statement [in the relevant language of statements]”
C) “There exists an x such that x * x = 1 + 1” is in the relevant language of statements
D) One cannot prove “There exists an x such that x * x = 1 + 1” from the list of axioms above
or
E) One cannot disprove "“There exists an x such that x * x = 1 + 1” from the list of axioms above
?