What he said. It’s clear that you’re not actually interested in learning anything at all, and that you’d rather just insult people who could help you. I really don’t have anything else to say.
Hawthorne
It look slike I was wrong to hold out that sliver of hope.
It looks like TVAA will go to his grave knowing nothing about GIT that cannot be discerned from a dictionary.
TVAA
That was a great example, BTW. It even showed that omega-consistent systems are a larger set than TVAA-consistent systems. Subtle and sweet.
The only place I’ve been able to find ultrafilter’s definition of “consistency” is from ultrafilter himself.
Every other definition, mathematical or otherwise, includes the idea that consistency involves freedom from contradiction.
Neither of you have demonstrated that you actually know anything about GIT… or what its implications are.
What you have done is repeatedly used a highly non-standard definition of a word in a general-language debate… a definition that no one here has actually been able to offer a cite for. ultrafilter’s “example” link concerned omega-consistency, not consistency.
And you accuse me of intellectual dishonesty? :wally
TVAA claimed:
That is certainly not true. The quotation from the link which I posted and ultrafilter clarified for me clearly defined consistency ( and went on to define [symbol]w[/symbol]-consistency).
You mean the bit that says K is consistent iff there is no formula S for which (let me see if I can translate this properly) S is in K and ~S is in K ?
Did I get that right?
I mean the bit that says that
“As usual K is consistent iff there is a formula S which is not a theorem of K or, equivalently, iff there is no formula S such that both S and ~S are theorems of K.”
At the very least, this refutes your claim that the definition being used by ultrafilter and Spiritus Mundi ( i.e. the first of these two) does not relate to consistency. ultrafilter went on to explain that, while these two definitions are equivalent in first-order predicate calculus, the same does not hold in more general contexts. In the absence of contrary evidence, I shall take his word for it since I know from past threads that he knows more about mathematical logic than I do.
I just read some old Phaedrus posts.
Oh. My. God.
Ah, but that’s just the point:
In plain old vanilla logic, an inconsistency means that it’s possible to prove anything. Therefore, if a system has a statement that it can’t prove, it must not contain any inconsistencies.
However, there are possible systems where an inconsistency does NOT mean that every possible conclusion can be drawn. In this case, the statement “there is a formula S which is not a theorem of K” is NOT equivalent to “there is no formula S such that both S and ~S are theorems of K”.
However, in such a system, I can see no logical reason for the first statement to be considered “consistency”, as it would then contradict the both the general meaning of the word and the meaning within standard logic.
It might be the case that some mathematical screw-up did name that property “consistency”… in which case both Spiritus and ultrafilter have screwed up, because their complaint is that I haven’t used the word “consistent” in accordance with its full mathematical meaning – when I’ve expressly said I am avoiding mathematical terminology.
ultrafilter has claimed that the first statement is the rigorous definition of consistency. But, since it applies to systems that contain contradictions (as he’s admitted), the word consistency is utterly inappropriate to describe this property.
Spiritus, I’ve been admiring you from afar for a loong time. Declare victory & call it a day (or a week or a month or something). Really. I know it feels like you’ve got a small dog that grabbed onto your pants cuff & won’t let go, but trust me: it’s a virtual dog. You won’t feel a thing if you ignore it.
I’ve been following this mess for some pages now. The problem here is obvious:
TVAA wrote:
In which case, you should expressly avoid claiming that your statements have any relation to GIT, which is properly expressed only in mathematical terms. Once you acknowledge that, you can go on to expressly avoid claiming that your half-baked hypotheses have any bearing on the real world (such as that GIT puts limitations on real computers), since you’re only talking about things which nobody else is talking about (TVAA-axioms, TVAA-logic, TVAA-GIT and TVAA-computers).
It’s entirely appropriate. You are just unwilling to learn what the word means in the context of rigorous mathematics. Your statement above is akin to saying, “blue is a color, therefore it is utterly inappropriate to use it to describe someone as being unhappy.” But once the context is clear, the meaning of someone “being blue” isn’t difficult to understand.
As should be obvious, your early assertions that one should be able to have a meaningful discussion about these concepts using common-language terminology have been shown, by you, through demonstration, to be completely wrong. The only way it’s possible is to slowly and painstakingly re-build the entire language of math through defition and context. But, Spiritus and ultrafilter have defined, over and over, how they are using the words they are using, and in what context they are using them. You are purposefully ignoring those definitions and that context, and so, in effect, you are simply refusing to see what they have been trying to show you.
Your willful ignorance is the reason this thread (and the GD thread) have dragged on for so long.
I declare both sides winners! Spiritus and ultrafilter win the rigorous math and logic side of the debates, and TVAA wins the TVAA-land version, which nobody cares about because it has no application to logic, math or the real world.
The WWW, unfortunately, does not provide many good resources for rigorous mathematics. I did find a site that has almost got it right. In its terminology:
[ul][li]absolute consistency: A system S is absolutely consistent iff at least ne wff of the formal language of S is not a theorem. [/li][li]simple consistency: A system is simply consistent iff there is no wff A such that both A and ~A are theorems. [/ul][/li]I have more usually seen what they call “simple consistency” referred to as “negation consistency”, but the two terms appear to be used interchangeably.
A better treatment is found (unfortunately not online) in Fundamentals of Mathematics, Volume 1, Behnke, Bachmannm, Fladt, and Suss.
“Algorithm” in the above is synonymous with “calculus” in the more rigorous sense and with “system” in the loose sense that we have been employing in this thread. Basically, a set of strings in a particular alhpabet along with rules of inference from which to derive new strings. Above, you can see that consistency is appropriately applied as a label for the generic quality: if some formula A is not derivable from the algorithm K, then K is consistent. Specifically, K is consistent with respect to A. If the formula A is bound to a particular string, then we determine one of the various “sub-properties” of consistency (omega-consistency, negation-consistency, etc.)
Really, I doubt there is any way to convince TVAA of this but that is the correct (rigorous) way to define “consistency” in mathematics. I understand that in many, many places you will see one of the “sub-properties” labeled simply as “consistency”. This is often a convenient thing to do and rarely causes misconceptions, but that is the right way to talk about “consistency” on contexts where it actually matters, such as noting that Godel’s original proof applies only to omega-consistent systems.
That said, as I was flipping through some dusty old books I began to think that TVAA might be right about one thing. Ultrafilter, the first place I recall seeing the statement that Godel-Rosser holds for generically consistent sets was when I came across erl’s old GD thread in which you posted a modern form of the theorem. I was intrigued and did some digging which seemed to support the idea that Godel-Rosser held for “absolutely consistent” systems. Cool!
But as I was tracking down a good treatment of “consistency” I came across Smullyan’s Godel’s Incompleteness Theorems, which has some pretty good treatments of the methods used by Godel, Tarski, Rosser, and Shepherson. As I flipped through the chapter on Rosser’s proof I noticed that Smullyan unfailingly specified the restrictive “simply consistent” which in his use is equivalent to “negation consistent”.
That made me backtrack through some of the historical overviews that I had used originally to confirm for myself that Godel-Rosser held for generically “consistent” systems, and I find myself questioning whether there might have been some terminology drift since the days of the Hilbert Programme. Methinks that what I was seeing referenced as “absolutely consistent” is what we would now properly call “negation consistent” or “simply consistent”. As I reviewed Smullyan’s proof of Godel-Rosser this view seems to be confirmed. I cannot be certain, though, since the source you pulled your treatment from might have a more powerful method of proof. Smullyan’s work is pretty good, but it’s still an introductory text aimed around the advanced undergrad/beginning gradschool level.
So, I’ll suspend judgment until I hear from you but it might be the case that while TVAA doesn’t know much about mathematical consistency he is correct about the scope of Godel-Rosser (though he persists in using the wrong “G” to invoke it.)
After reading (well, skimming through) both TVAA’s and Spiritus Mundi’s arguments, I realize that I want to have sex with them. Both of them. At the same time, if possible. Did I mention I have a geek fetish?
We now return you to your regularly scheduled pitting.
In any system that doesn’t contain A and ~A, there will be statements it doesn’t contain. That is obvious. But not containing all statements does not necessarily that A and ~A aren’t included for all values of A.
Actual consistency is a subset of SM & ultrafilter “consistency”. In every source I have managed to locate, ‘consistency’ is always used to indicate “freedom from contradiction”, whether we’re talking about standard language or mathematics. The SMU definition does not imply this in all cases.
Without a cite demonstrating that there are indeed branches of mathematics that use the word ‘consistency’ in such an inconsistent and counter-intuitive way, I’m gonna hafta conclude that you two are full of it.
Oh, and DaveW: the whole point is that GIT can be expressed in normal language. Your conclusions therefore do not follow from reality. Thank you for playing.
TVAA wrote:
You have yet to demonstrate that you can, indeed, express today’s GIT in normal language. That is part of my point - that you have repeatedly failed to make your point. Your conclusion that my conclusions don’t follow from reality is, itself, unfounded and premature until you demonstrate your ability to do as you claim to be able to do.
This isn’t a “hafta”, it’s a “wanna”.
Consistency is a formal property and is defined quite generally. When one constrains the formula which cannot be derived, one specifies a specific type of consistency. I have provided a citation to exactly that effect. Apparently, you just plan to ignore it because it is not available in hypertext. shrug
Strangely enough, you didn’t seem to require that all acceptable references be in electronic format when reaching for your introductory logic text (or when telling us all about your “invisible experts[sup]Tm[/sup]” for that matter.)
Goodness, who could have predicted such a turn of events.
:rolleyes:
I had plenty of other sources that were readily accessible. (And the only thing I’ve ever tried to convince any one of with my “invisible experts” is you of the fact that you really don’t understand the concepts you’re throwing around so readily.)
“Specific type of consistency”?! Consistency doesn’t constrain a formula, it constraints lots and lots of them.
Spiritus had said [bracketed words my reading]:
As far as I can tell, nothing was said about consistency either constraining or not constraining a formula – rather, what was being said was that when you [a person] decide to manipulate equations in such a manner that a particular formula cannot be derived from those equations you [a person] specify a certain type of consistency.
and of plain English, too
So what? You offered information from sources that are not available online. Did you do so under teh presumption that information not available on the Internet is somehow “tainted”? Do you imagine that every item of human knowledge will appear in front of you if you type the proper string into google?
Yes. Do you honestly still not understand that there is more than one type of consistency? Have you bothered to click links or read sites as they have been offered?
This is pathetic. I don’t recall the last time I encountered anyone quite so dedicated to preserving his personal ignorance.
Or one who was so dedicated to misinterpreting or misrepresenting simple statements. What I said was: When one constrains the formula which cannot be derived, one specifies a specific type of consistency.
[ul][li]Consistency is a property of a calculus (or a more complex deductive system).[/li][li]Consistency is present when at least one systactically legal formula in the calculus cannot be derived.[/li][li]Specifying a particular formula which cannot be derived, such as (A & ~A), identifies a particular type of consistency. [/li][li]Consistency does not constrain formulas. Consistency is simply a description of a property of a calculus (or more complex deductive system).[/li][li]Fewl, you got it right except that the choice is not so much about how to manipulate equations (the choice of system.calculus specifies that) but about what property to look for/evaluate.[/ul][/li]
Since you seem to want to cling to the delusion that “consistency” can mean only one thing, here is a link (gosh, someone seaid it on the web so maybe it’s true) to a glossary of terms from logical systems. Notce the entry for:
This site mentions both absolute and simple consistency but fails to show that “simple” is just a special case of “absolute”.
Happy now?
“Absolute consistency” is much, much weaker than “simple consistency”.
(i) always implies (ii), but (ii) does not always imply (i).
So why should we care about (ii) in regards to Godel Incompleteness? It’s trivial to show that a system doesn’t contain the proof of a false statement… GIT is important because it shows that some statements can be neither proven nor refuted.
You haven’t been talking about “absolute consistency”, which is a requirement for true “consistency”. Even bringing up omega-consistency is just a red herring (isn’t lowercase-omega generally used in physics and math to denote some variety of ‘weakness’?)… You claimed that was the definition of “consistency” without qualifiers.
Your statement is incorrect. Admit it, and let’s move on.