.999 = 1?

[Indistinguishable]

@septimus

In other words, it’s a Reductio ad Absurdem demonstrating that the Axiom of Choice must be false. :eek: It’s similar to refuting the “obvious” fact that deterministic games are determined.

What’s so “obvious” about the Axiom of Determinacy? Perhaps the jargon (in which different concepts of “deterministic” and “determined” are given similar names) is misleading. Of course, there are plenty of games which are “deterministic” but not “determined” [e.g., rock-paper-scissors: deterministic in that the outcome is determined entirely by the players’ choices, but non-determined, in that neither player can unilaterally prevent any particular outcome].

Let’s write out what the Axiom of Determinacy amounts to: By the “children” of a finite bitstring, I mean both ways of adding a single bit to the end of it. By a “fully determined game”, I mean a function from finite bitstrings to {happy, sad} such that a bitstring is considered happy if any of its children are sad, but considered sad if both of its children are happy. Relative to a fully determined game, a countably infinite bitsequence is considered “well-played” if its finite prefixes eventually alternate between happy and sad; we will call a well-played sequence “even-happy” or “odd-happy” according as to whether it is the even- or odd-length prefixes which are eventually constantly happy.

The Axiom of Determinacy says that for every function from countably infinite bitsequences to {even-happy, odd-happy}, there is a unique “fully determined game” agreeing with it on all well-played sequences.

Why should we expect this?

[Indistinguishable]

Conversely, the Axiom of Choice might be framed this way:

Suppose there is an island [always an island or a prison…] with many islanders on it [possibly infinitely many!], and each islander has at least one hair. The islanders are reliably informed that the vengeful math-gods will annihilate them all unless each plucks a single hair from their head. The Axiom of Choice says the islanders can avoid annihilation; the denial of the Axiom of Choice is to state that the islanders are doomed.

“Of course, each islander will just pick some hair and pluck it!”. Might this not just as well be considered a reduction ad absurdum of the denial of the Axiom of Choice?

[Indistinguishable]

But I would not want to speak of these things as though there’s some external truth of the matter. That’s ridiculous; as ridiculous as wondering whether kings can “truly” capture pieces by jumping over them, or whether multiplication “truly” is commutative (or whether infinitesimals “truly” exist, for that matter). There are abstract games we find interesting* in which it’s one way, and abstract games we find interesting* in which it’s another way (even abstract games in which we decline to take a position on the matter). Yes, we happen to use similar language in different mathematical stories, because they still do have similarities, but nonetheless, any unsettling sense of conflict is a mirage. There’s no conflict or external facts to settle because we make up the rules instead of discovering them, electing to study this or that preformal concept with this or that formal system.

Talking about the objective truth of the matter is like talking about the objective truth of Sherlock Holmes’ blood type. There isn’t one; explore whatever fan-fiction you like. We may rally sometimes around a canon, but even that is just a caprice of history and convention; if you want to consider the idea of Sherlock Holmes living in the 21st century, or Watson being a woman, or any other such thing, knock yourself out.

[*: Or useful, or accurate as a description of some phenomenon through some interpretational bridge, or what have you]

[septimus]

@Indistinguishable

Of course, there are plenty of games which are “deterministic” but not “determined”

…
Why should we expect this?

I’m using “deterministic” to exclude games where a player cannot determine his opponent’s (pure) strategy. Note that perfect-information turn-taking games are fundamentally different from simultaneous-play games like Rock-Paper-Scissors.

The relevant determinancy is trivially true for finite games. True, infinities can stagger the imagination but extending the determinancy to infinite games seems far more “plausible” to me than Itself’s almost unbelievable claim. YMMV.

[septimus]

@Indistinguishable

“Of course, each islander will just pick some hair and pluck it!”. Might this not just as well be considered a reduction ad absurdum of the denial of the Axiom of Choice?

That’s like saying that displaying matrices AB=BA “proves” matrix multiplication always commutes.

… And you better drink another cup of coffee if you really wonder about Choice over a finite set like head hairs.

[Indistinguishable]

@septimus

That’s like saying that displaying matrices AB=BA “proves” matrix multiplication always commutes.

I don’t follow you; what do you mean?

[Indistinguishable]

@septimus

The relevant determinancy is trivially true for finite games. True, infinities can stagger the imagination but extending the determinancy to infinite games seems far more “plausible” to me than Itself’s almost unbelievable claim. YMMV.

What of the person who says “The relevant choiceability is trivially true for finite indexes. True, infinities can stagger the imagination but extending the choiceability to infinite indexes seems far more ‘plausible’ to me than septimus’s almost unbelievable claim.” You, indeed, are free not to care about such a way of looking at things, but this is in fact, how most mathematicians in practice think about it. I wouldn’t agree with them either, but since the argument you give is the same as your opponents would, it’s not in itself a very compelling reductio [then again, I suppose the term “reductio” often connotes some kind of preaching to the choir].

Lots of things are true of the finite and false of the infinite. Why should choice be one of them and determinacy not, rather than the other way around? Well, you can restrict attention to whatever you want, of course.

Here’s a fact that’s true about finite games: Suppose the game is such that, on every turn, I can either forfeit or make another move, I always have at least one move available to me other than to forfeit, and I can only lose the game by choosing at some point to forfeit. Then I can guarantee a win for myself, trivially.

Is this true about infinite games? Thinking of each turn as a choice to be made, this amounts to the Axiom of Choice; in the absence of Choice, it may not even be possible for me to win, much less force one.

[septimus]

@Indistinguishable

I don’t follow you; what do you mean?

You posit a case where Choice is easy (on finite sets no less :smack: ) and then suggest it is a “reduction ad absurdum of the denial of the Axiom of Choice.” In context the double-negative “reduction ad absurdum of the denial of …” implies a proof. As though an instance of AB=BA were a “reduction ad absurdum of the denial” that multiplication doesn’t always commute.

[septimus]

@Indistinguishable

You, indeed, are free not to care about such a way of looking at things, but this is in fact, how most mathematicians in practice think about it.

I might find one of the hypotheticals more or less “plausible” than the other, but if I were a Real Mathmatician like you I’d either find them equally plausible/implausible or ignore palusibility altogether. Thanks. Still the dear old SDMB we all love so well.

[Indistinguishable]

@septimus

… And you better drink another cup of coffee if you really wonder about Choice over a finite set like head hairs.

Why not? In the absence of Choice, even if every islander has exactly two hairs, we aren’t guaranteed that each can pick one to pluck. The relevant infinitude is the number of islanders, not each islander’s number of hairs.

[schooner26]

@Great_Antibob

See, you’re still not getting it.

If you are going to talk like this, then please don’t address me anymore.
I “got” it fine during the several engineering calculus classes I took.

“Belief” has nothing to do with it.

Sure it does. Mathematicians collectively “agree” on certain things.

That’s the great (and often confounding) thing about math. Once you set your definitions, what follows requires only rigorously following through the logic.

There is nothing logical about completing an infinite subtraction. It is oxymoronic to say “completing an infinite operation”

What’s happening is you don’t like the definitions but you still want to keep the conclusions.

What ? No. Infinite series are not numbers. Any number is an approximation of the infinite series based on an acceptable precision.
Endless 9’s in 0.(9) are simply endless 9’s. Can it describe any amount ? Of course not. You can voluntarily take the limit, but doing a math operation on some object, which has a result, doesn’t automatically equal the original object before the operation was performed !!

(even if you don’t accept/understand them).

I am beginning to think that you are the one who doesn’t actually understand them. You can assert them, and state them, but that is all I am seeing.

No remainder exists. You claim otherwise, but you’re wrong.

No I am correct. A remainder must exist in order to have another digit in the quotient. Both exist simultaneously.
So again, *you *are the one who is wrong.

at odds with how they have been defined to work.

Defined to work, doesn’t mean they are correct.
Attempting to represent the sequence as a decimal number is now how it is constructed. The constructed number is output after analysis is done on the sequence and its partial sums.

Is the number 0.875… equal to 1 as well ? That is a representation of the sum of 3 terms in a sequence which converges on 1.

Just because you have an infinite series doesn’t mean you require infinite time or infinite computation to solve.

So infinite is not endless ? :smack: Are you implying that you *can *complete the process ??

    x =  sum (i = 0 to inf) g*

where g* = 2 if i = 0 and g* = 0 else.

You are using an analogy or example of 0s. Zeros are a special in almost every number system. You are trying to convince me that adding nothing is the same as adding something.
Try again.

 1 - [ sum(i = 0 to inf) 1/2^i]

This is equivalent to sum (i = 1 to inf) 1/2^i, since 1/2^0 = 1.

You are assuming the limit in your process.
An infinite series is a complete object. Although you think you can just separate individual terms out, you are breaking the infinite series theorems.
That term is not what you think it is. It is the limit of another infinite series.
It is obvious you haven’t explored this area before.

other than you and some other folks who are very wrong

Not wrong at all. There is nothing magical “at infinity” that causes a carry so that all the 9’s roll over to 0’s, giving 1.000…
In order to prove the infinite sum equals the limit, you need to calculate the infinite sum, which cannot be done. Therefore, you are just defining it, making it up. Nothing is proven. Therefore, I am not “wrong” as you assert so well.

If you begin by assuming two different decimal representations must represent different values, of course you will conclude they have different values. You are assuming your conclusion.

No. I am assuming a decimal number represents a single value, the whole purpose of having numbers.

Real numbers have one value. But they can have multiple decimal representations.

A flawed system indeed.

A decimal representation is NOT the same thing as the value of the number. It is, as seen from the words themselves, a “representation” of that value.

I am assuming that this is how someone convinced you to believe.

There are numerous things which have multiple representations. 1/2 = 2/4 is a pretty simple one.

Not an accurate analogy whatsoever.
I was waiting for the “infinite fractions having the same value” failed analogy.

it’s quite possible for a single number to have no unique decimal representation.

Close. But I understand your point, and this is my point too. Some “objects” or values cannot be represented by a decimal number !!
Sqrt(2) for instance.
However, I say there is no decimal number that can represent it. Not “decimal representation”

Appealing to Euler and ignoring Cantor means you don’t have to consider how mathematicians actually think about the real numbers. It’s quite relevant and ignoring it means you’re dealing with a strawman representation of what mathematicians claim.

Saying it is strawman, when it is not, is strawman. Well done.

[schooner26]

@Buck_Godot

but if 0.999… is going to have any well defined meaning it is going to have to equal 1.

Why does it need to have a value ? Just because I write a bunch of digits down, doesn’t mean it must have a value.

same way they understand that Sigma sum notation with an upper limit of infinity represents the limits of the partial sums.

I disagree with that also.

These have different meanings:

Σ (n=1 to ∞) 9/10ⁿ = 0.999…

lim Σ (n=1 to N) 9/10ⁿ = lim ( a - ar^N) / (1 - r) = a / (1 - r) = 1
N→∞

2 totally different concepts. First one is an infinite string of numbers. Second is the limit concept.

[Great_Antibob]

@schooner26

Sure it does. Mathematicians collectively “agree” on certain things.

They agree on definitions and starting points. After that, they don’t “agree” on things - they prove them.

“Belief” still doesn’t come into it. As we have now repeated ad nauseum, it’s ok to use a different set of definitions and starting points, as you are attempting to do. But you can’t take one starting set and hope to modify it after you have some results.

In this case, you don’t like the implications of this system. As I noted earlier, it’s akin to earlier attempts to prove the Parallel Postulate. Many “proofs” consisted of exactly the same sort of ‘proof by incredulity’ you’d like to employ here.

@schooner26

Is the number 0.875… equal to 1 as well ? That is a representation of the sum of 3 terms in a sequence which converges on 1.

So, are you saying that as Buck Godot claims, your problem is with the notation?

For what it’s worth, the number does not converge to 1. The difference between 1 and it is pretty big (bigger than 1/10).

For 0.999… and 1, there is no non-zero real number that can be found to be the difference. If there is a difference, it has to be smaller than every other real number, except 0.

And, yes, that can lead to alternate number systems, including ones with infinitesimals, as mentioned repeatedly in this thread. But it’s not the standard real number system.

@schooner26

You are using an analogy or example of 0s. Zeros are a special in almost every number system. You are trying to convince me that adding nothing is the same as adding something.

Then, please be precise.

This isn’t a mathematician/engineer thing (and yes, I do have degrees in both). In both engineering and pure mathematics, how precisely you specify things does matter.

You initially stated that you can not “solve a finite number minus an infinite series”. Those were your words.

It seems you meant that you actually can solve them in some cases but not all of them. What cases? When do you claim it is not possible? Elucidate, please.

This isn’t me being pedantic. It’s me trying to understand what precisely you are claiming, as ambiguous/imprecise statements will get us nowhere.

@schooner26

No. I am assuming a decimal number represents a single value, the whole purpose of having numbers.

Well, why do you assume it?

By the way, it’s fair to make this assumption. Others have done so, e.g. systems with infinitesimals. But it doesn’t lead to the real number system. Nor does the existing standard real number system work properly if you try to force it in like that.

@schooner26

Sqrt(2) for instance.
However, I say there is no decimal number that can represent it. Not “decimal representation”

Nonstandard use of both terms. Again, you can use/define terms as you wish. But that doesn’t make them make sense in the standard real number system.

To explain, there are just numbers. You can write a number in a multitude of ways.

Under our standard number, a “decimal number” as you put it doesn’t exist as a concept. You can write numbers however you want. It just happens that in the way we typically choose to express numbers, representations of numbers aren’t unique.

For example, 1/3 can be written just fine as 0.1 in base three. Just changing our number base turns a number with infinite decimal representation to a number with a finite decimal representation. The number itself doesn’t change - only its decimal representation. But even in base three, there are numbers with multiple decimal representations.

Does that means whether or not the quantity 1/3 can even have a decimal representation is dependent on our use of number base? That doesn’t make sense. The way we set up the reals is independent of number base.

Representations are just that - representations. The numbers themselves are just fine, even if there can be two or more representations of them.

[Zakalwe]

@Great_Antibob

Representations are just that - representations. The numbers themselves are just fine, even if there can be two or more representations of them.

Actually, I heard the numbers called in sick today - something about a late night with the letters. Numbers never have been able to hold their liquor worth a damn.

[7777777]

@watchwolf49

What is 1 - 0.999… equal to?

It is equal to infinitesimal x.

There is the same infinitesimal difference between infinity and the limit of infinity.

The value of the limit of infinity is not equal to infinity, a simple proof is that infinity
is not a number whereas the value of the limit of infinity is a number.

I know, many people here try to deny what I say, but because they don’t know
what number is the value of the limit of infinity, they don’t sound very convincing.

the value of the limit of infinity ≈ ∞
because of the infinitesimal difference x

Also
1 ≈ 0.999999…
because of the infinitesimal difference x

[Frylock]

Do you mean the value of x as x reaches infinity?

I’m not sure what the phrase “the limit of infinity” is meant to mean…

[Frylock]

I haven’t followed any of these recent conversations, but while I know there are plenty of different contexts in which the word “infinity” might refer to different things, I was taught a concept of infinity according to which it is a number. It’s not a natural number or a real number, but an infinite number. It’s defined as the smallest number which is greater than every counting number.

Numbers are sets, each number being the set of all the numbers less than it, and infinity being the set of all counting numbers.

So anyway, I know that solves nothing.

[Indistinguishable]

Yes, that’s a wonderful concept. So far as high-falutin’ jargon goes, such numbers are called “ordinal numbers”, their representations as sets are called “von Neumann ordinals”, and the particular number matching your concept of infinity (i.e., the smallest transfinite ordinal) also frequently goes by the name “ω”.

[watchwolf49]

@7777777

@watchwolf49

What is 1 - 0.999… equal to?

It is equal to infinitesimal x.

Infinitesimal x is not an element of the real numbers. Both 1 and 0.999… are. Since reals are closed under addition by definition, the difference between 1 and 0.999… must be a real number.

What is this real number?

[schooner26]

@Great_Antibob

For what it’s worth, the number does not converge to 1. The difference between 1 and it is pretty big (bigger than 1/10).

No, 0.875… is an infinite series, “represented” as the sum of the 1st 3 terms of the series, just as 0.999… is the sum of the 1st 3 terms.

1/2 + 1/4 + 1/8 + … = 0.875… = 1

For 0.999… and 1, there is no non-zero real number that can be found to be the difference.

No number can be found because you are asking a question which makes no sense. “What is the difference between a finite number and an infinite series?”
Thus, the answer which tries to appear is 1/10ⁿ
1 - 0.9(n-times) = 1/10ⁿ

you allow n=∞ for 0.9(∞ times) but do not allow n=∞ for 1/10^∞
That is where I step in and say you can’t have it both ways.
This is where 1/10^∞ is not a number as a result of 1 - 0.(9), and we know 1 is a number, leading to a conclusion that 0.(9) is not a number either.

If there is a difference, it has to be smaller than every other real number, except 0.

Has to be, or “is” ?

You initially stated that you can not “solve a finite number minus an infinite series”. Those were your words.

And I shall stick by them. 1 - 0.(9) is like asking “what is 2 apples - 1 orange?”

It seems you meant that you actually can solve them in some cases but not all of them. What cases? When do you claim it is not possible? Elucidate, please.

Not sure what you are talking about here.

This isn’t me being pedantic. It’s me trying to understand what precisely you are claiming, as ambiguous/imprecise statements will get us nowhere.

I have said it a few times very clearly.
The items used in constructing 1 are not equal to 1.

1 is output as a result of construction.
Again, the sequence 1/2 + 1/4 + 1/8 + … “equals” 1 (according you). So how do I 1/2 + 1/4 + 1/8 + … in a similar manner to 9/10 + 9/100 + … ??
0.875… = 1 ??

Representations are just that - representations. The numbers themselves are just fine, even if there can be two or more representations of them.

Can you define the differences between numbers and representations and any other unique terms you are using ?