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Mickey1
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A real circus
« on: Jun 23rd, 2013, 6:56am » |
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Assume one can be aware of “all” the numbers of an infinite number series (perhaps by divine intervention, or by accepting a divine axiom) such as defined by a zero followed by a comma, followed in turn by an infinite amount of numbers.
For lack of a better name, let’s call them real numbers (in an interval). We don’t really know what they are and we are not sure about the interval’s upper bound.
We can now take such any such number and make a series by truncating it after n decimals, then after n+1 decimals, after n+3 decimals etc. We have thus found a method of defining real numbers by Cauchy sequences. Observe that the first step, the truncation, was possible since you knew all the infinite series’ elements (i.e. all the real numbers in the interval) in advance.
You might recall that the last part of Cantor’s diagonal argument is reached by identifying an infinite series of numbers and referencing - not a series - but the actual divine number with its infinite decimals all considered one by one, from number one to infinity.
You might then consider the often heard question: What is the difference between i) 1 and ii) 0.9999…
It is easy to answer. The first, i), is a number, the second, ii), is a zero followed by a (decimal?) point, followed by four nines and three dots. However, we are prepared to make a good faith effort here, and interpret ii) the following way: it is a string of symbols that lead us to conclude that it is actually a plea from the writer for us to replace it with something more “number-like”.
One explanation is that it is related to a sequence such as 1-10^(-n), n= 1,2,3,4,… The series have the elements 0.9, 0.99, 0.999, and it also has a limit, 1, as n goes to infinity. The writer might therefore also have had either the series, or the series’ limit =1 in mind, or perhaps both?
In any case, in his proof, Cantor needs them to be different (1 and 0.9999... that is), or he would be up in Diagon Alley.
Being devine-powered, we also have a third alternative. The writer is one of us and ii) is made up of a zero followed by an infinite number of 9s. We have that number in our reference bank, unlike the mere mortals. This number is obviously different from the number 1 followed by a decimal point, followed by an infinite number of zeros. This brings into question the numerical value of infinite- or divine-defined numbers and makes us conclude that they do not have a one-to-one relationship with the real numbers as understood by mortals. I hope you can follow me in this religious labyrinth. We also see now, that the divine intervention looks more and more like a deal with the devil.
Another observation is that (divine and naively) 0.9999… seems to be 3 *0.3333… , but 0.3333… is 0.1 in base 3. This makes us think that there is another number 0.02222… in base 3, which is different from 0.1 also in base 3. We have now several numbers: the first A=1, the second B=0.9999.. (divine) and C=3*0.02222… (base 3 and divine). We would expect that A>B >C. All different but they might have the same numerical value if you ask the mortals. (Edit: I am not really sure about C since I use the relation 1/3=0.33333..)
I could go on but I believe the reader is already exhausted.
I accept mortal comments on this.
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| « Last Edit: Jun 23rd, 2013, 12:05pm by Mickey1 » |
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rmsgrey
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Re: A real circus
« Reply #1 on: Jun 24th, 2013, 5:33am » |
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I've not read Cantor's original proof, but it's easy enough to produce a version of the diagonalisation argument that isn't sunk by an infinite string of 9s - when creating the new number, set the nth digit to be 4 except where the nth digit of the nth number is 4, then use 5.
Since the new number is entirely made up of 4s and 5s, it cannot end with an infinite string of 0s nor an infinite string of 9s, so cannot be the same as a number that looks different.
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"1" is obviously different from "one" - the first is a single numeral; the latter is a sequence of letters. Unless you consider them to be different numbers, you need to provide more explanation of why you consider 1.000... and 0.999... to be different numbers than that they're written differently.
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Mickey1
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Re: A real circus
« Reply #2 on: Jun 24th, 2013, 11:38am » |
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I note you propose a “mortal” argument which I accept fully.
My point about Cantor is not that he should avoid a lot of nines and use your trick (which I assume he may have done). It is perhaps better explained the following way:
A diagonal for a finite version of the exercise is reached when you arrive at the other. diagonal, corner. However for an infinite series of numbers, any Cantor-style diagonal argument (whether it is about 4’s, 5’s or other none-9’s) must use a number 0.abc… with an infinite “number of numbers” admittedly with a possible lot of 4s, which is what I call a divine concept.
Given this concept there is no point in asking what 0.9999… (interpreted as in an infinite number of 9’s) is, it is just exactly those infinite numbers of nines. And 1 or 1.000… is clearly something else.
Once settled in an infinite environment as I define, the infinite numbers need no interpretation. Some numbers, definable, can be replaced by a formula that produces them. These formulas can be Gödel-enumerated and arranged in an ordered sequence including 0.9999….
Not so Cantors number which cannot equal to any other number in a countable set, the proof coming through “at infinity” and no sooner.
The whole problem is therefore all about definability, a notion very few people like, (“this question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form” used as argument for closing my question on math.exchange with minus point as penalty) although I don’t really understand why.
Definability has its Wikipedia entry, to which Joel Hamkins comments in http://mathoverflow.net/questions/44102/
“ although seemingly easy to reason with at first, is actually laden with subtle metamathematical dangers to which both your question and the Wikipedia article to which you link fall prey. In particular, the Wikipedia article contains a number of fundamental errors and false claims about this concept.
It seems I am not alone in my confusion.
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rmsgrey
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Re: A real circus
« Reply #3 on: Jun 25th, 2013, 3:35am » |
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on Jun 24th, 2013, 11:38am, Mickey1 wrote:| Not so Cantors number which cannot equal to any other number in a countable set, the proof coming through “at infinity” and no sooner. |
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Your phrasing here is poor - there is no one "Cantors number" - what you get by applying the diagonalisation argument to a specific countable set in a specific order is a number specific to that order of that set - other countable sets and other orderings may give other numbers. In particular, you can add the number generated by a given set to that set to create a new, still countable, set, with an extended order, which will generate a different number.
on Jun 24th, 2013, 11:38am, Mickey1 wrote:| Given this concept there is no point in asking what 0.9999… (interpreted as in an infinite number of 9’s) is, it is just exactly those infinite numbers of nines. And 1 or 1.000… is clearly something else. |
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You've just given three different strings, and claimed that one of them is clearly different from the other two, in a way that implies that you think that the other two strings are equivalent in some way.
Why is it "clear" that 0.999... is something other than 1.000... but not that 1 is something other than 1.000... nor that .999... is something other than 0.999...? You'll have to do better than saying "it's obvious" - that's not a proof; it's a claim that something doesn't need to be proven.
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Mickey1
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Re: A real circus
« Reply #4 on: Jun 25th, 2013, 4:48pm » |
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About many "Cantor numbers"
Let me first say that there would be a “Cantor number” i.e. a number constituting the proof in any diagonalization argument for a specific countable set of numbers – these numbers would all be infinite. There would also be an infinite number of examples of numbers which all would provide the proof if we use base n, where n>2. For n=2 there might be only one counterexample (this leads perhaps to a somewhat different issue, where by the way it would be more difficult to design the trick to avoid 0,1111... base 2 ).
An experiment could be done using pi and a numbered arrangement of the rational numbers between 0 and 1. In such an experiment, we would have an infinite series of numbers we could point to (i.e. those belonging to pi’s decimal expansion) and explain that this number would not be among the countable set, but we would not necessarily have to know the infinite diagonal number. Instead we would have access to a number of series (i.e. formulas) for its (pi’s) calculation.
However, any arbitrary enumeration of ALL countable numbers would, through Cantor’s proof process, lead to a Cantor-diagonal number (numbers) which we would not be able to calculate through a formula, since the formulas can be (Gödel-) enumerated and have a one-to-one relationship with our enumerated number series. The translated formula-series would (perhaps) be different from the series we started out with, but the diagonalization would show that no such formula (using a finite number of instructions) can exist which calculates the Cantor diagonal infinite number. We would have to “memorize” an infinite number as the only option of knowing it. These are the undefinable real numbers par excellence (which nobody seems to like).
About the different numbers 1, 1.000… and 0.9999… I consider all numbers as infinite here so that 1=1.000… so the only remaining question is about 1.000… and 0.9999….
By “clearly” different I mean that a diagonalization-like activity (robot-like) encountering 0,9999 would say first: 1 is different from 0 (referring to the first digits), followed by an infinite series of statements: 9 is different from 0, 9 is different from 0, 9 is different from 0…
This is I believe how Cantor’s argument works, and my infinite-capable robot or PC would consider the two numbers as different. I believe that this exhausts both my understanding of the real numbers and my semantic abilities.
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0.999...
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Re: A real circus
« Reply #5 on: Jun 25th, 2013, 5:27pm » |
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on Jun 25th, 2013, 4:48pm, Mickey1 wrote:| However, any arbitrary enumeration of ALL countable numbers would, through Cantor’s proof process, lead to a Cantor-diagonal number (numbers) which we would not be able to calculate through a formula, since the formulas can be (Gödel-) enumerated and have a one-to-one relationship with our enumerated number series. |
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Could you rephrase your first sentence here? It is suggesting that some (real) numbers are countable, which is a notion I have never come across.
on Jun 25th, 2013, 4:48pm, Mickey1 wrote:| By “clearly” different I mean that a diagonalization-like activity (robot-like) encountering 0,9999 would say first: 1 is different from 0 (referring to the first digits), followed by an infinite series of statements: 9 is different from 0, 9 is different from 0, 9 is different from 0… |
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The only reason I see this distinction coming about in diagonalization has already been handled by such a modification as the one rmsgrey suggested. In other words, it would be erroneous to use the diagonalization process in a form which distinguishes 0.999... and 1.000... and the like. To elaborate on why it should be avoided, infinite sums defining them are Cauchy sequences of rationals, so the differences of their partial sums can be measured by rational numbers. In this case the sequence of differences has a limit among the rational numbers, 0. Notice that a real number was never actually constructed in this process as one can work with Cauchy sequences of rational numbers without working with real numbers.
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| « Last Edit: Jun 25th, 2013, 5:50pm by 0.999... » |
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Mickey1
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Re: A real circus
« Reply #6 on: Jun 26th, 2013, 5:04am » |
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It is appropriate for you to comment, given your name.
We are back in the issue of constructivity. Some real numbers must exist which can only be constructed by Chauchy sequence if you knew them in advance or had a formula to calculate them.
Alternatively, there are some real numbers for which the Cauchy sequence (having those real numbers as limit) cannot be defined by a formula in a finite number of steps. I have a feeling this was what Kronecker objected to in Cantor’s work.
Some say this is related to the axiom of choice, but I wouldn't know (out of ignorance, being an amateur). It does follow from my view that there are real numbers that cannot be chosen if by chosen you mean referenced by a finite formula. I am not sure that is the same thing.
You may still accept numbers which cannot be defined (by a finite formula), including those we consider to be all the real numbers. The geometric notion of a continous line is quite suggesting. All the same, I note there are also quite a few problems surronding the continuum.
Let us also recall Prof. Hamkins about undefinable numbers once more: “although seemingly easy to reason with at first, is actually laden with subtle metamathematical dangers to which both your question and the Wikipedia article to which you link fall prey. In particular, the Wikipedia article contains a number of fundamental errors and false claims about this concept”.
We see that there are some issues here.
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0.999...
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Re: A real circus
« Reply #7 on: Jun 26th, 2013, 7:08pm » |
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I am sorry, but I do not see how any of that addresses my point. In a nutshell, 1.000... and 0.9999... can be shown to be indistinguishable as limits of Cauchy sequences using a number system that (as far as I can tell) you accept exists, namely the rationals.
Technically, we are using the metric space of rationals with a metric whose target is the set of rational numbers.
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| « Last Edit: Jun 26th, 2013, 7:30pm by 0.999... » |
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rmsgrey
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Re: A real circus
« Reply #8 on: Jun 27th, 2013, 4:35am » |
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"The smallest positive integer which cannot be uniquely identified in fourteen words or fewer" looks like the definition of a number (with a large but finite vocabulary, there is a quite large, but still finite number of possible strings of fourteen words or fewer, of which the vast majority will not uniquely identify numbers, some will uniquely identify numbers which aren't positive integers, and only a tiny fraction (though still large in absolute terms) will uniquely identify positive integers. Once you've listed all the strings and crossed off all the positive integers that are uniquely identified, you will still have infinitely many positive integers left, one of which will be the smallest.
The trouble is, since the string used to describe this number is only fourteen words, it's self-contradictory.
You don't need the reals to get numbers without a finite definition - the vast majority of positive integers are undefinable too.
The axiom of choice says that, given any collection of non-empty sets, you can always form a new set by picking an arbitrary element from each of those sets - for example, you could (ignoring logistical difficulties) run a lottery where each number is picked from a new set of balls by a new machine (maybe we could get Hilbert's Hotel to host it?) - if each machine holds ten balls, then the winning numbers can be naturally interpreted as a decimal representation of a real number - letting you pick an arbitrary real number.
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0.999...
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Re: A real circus
« Reply #9 on: Jun 27th, 2013, 5:46am » |
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on Jun 27th, 2013, 4:35am, rmsgrey wrote:| You don't need the reals to get numbers without a finite definition - the vast majority of positive integers are undefinable too. |
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(A very good post, and I am just nitpicking.) I was under the impression that the usual recursive definition suffices here.
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rmsgrey
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Re: A real circus
« Reply #10 on: Jun 28th, 2013, 6:36am » |
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on Jun 27th, 2013, 5:46am, 0.999... wrote:
(A very good post, and I am just nitpicking.) I was under the impression that the usual recursive definition suffices here.
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Blech. You're right - every finite integer (which is all of them) has an unlimited number of finite, but long definitions - got my "for all" and "there exists" the wrong way round in "for all n, there exists a number m such that m cannot be defined by a string of length n or less"
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Mickey1
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Re: A real circus
« Reply #11 on: Jun 30th, 2013, 2:20pm » |
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I would like to comment later and take a small vacation from the numbers 0.9999… vs 1.0000… because these are both numbers definable in a finite number of steps. It is the real numbers’ definition which is the problem. They are only interesting in that they address an infinite decimal expansion as if this was without problems.
Let us now define real numbers between 0 and 1 in the following matter using base 10: We start with the following arrangement: A zero followed by a comma, as we would understand it from the rational numbers’ decimal representation, followed by an infinite number of bins each of which has the content defined by the numbers 0-9. A particular real number is then a special case of this arrangement where a particular number is chosen for every bin. This is one of Wikipedia’s several definitions: “or certain infinite ‘decimal representations’”.
I took the opportunity to read Wikipedia’s definition of the axiom of choice, the informal formulation (hoping it is valid in spite being informal): "Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin".
This is equivalent to saying using my (and Wikipedia’s third ) definition that one of my real numbers can be chosen (i.e. defined or referenced). This means that the axiom is not really an axiom. It is just a definition of otherwise non-referenceable real numbers. It seems hare-brained to call it an axiom. The Dedekind cut ofthe Cauchy series will only define a countable subset of R unless an infinite long formulations are involved.
Of course you need to get the timeline right to avoid an ahistoric view: I assume people had a naive conception of the real numbers in the absence of a strict definition, and then ran into the need for an “axiom” as a sin of omission.
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0.999...
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Re: A real circus
« Reply #12 on: Jun 30th, 2013, 8:40pm » |
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The axiom of choice is an extremely powerful statement, and your proposed equivalence is not at all a logical equivalence.
In fact, we do not even need a weak form of the axiom of choice to determine a real number following the scheme you gave: we can just select 0 from each "bin" to get a perfectly valid choice function. The axiom of choice is not needed to construct that; the other axioms of set theory will do.
The other main misconception I see in that post is that the axiom of choice constructs everything for you. No, it merely guarantees the existence of just one choice function for a family of sets.
Thus, "[the axiom of choice] is just a definition of otherwise non-referenceable real numbers," is a false statement for both reasons given above.
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Without looking at references, I would guess that the use of the axiom of choice as it relates to definability of real numbers is significantly more subtle, and I would guess that this topic lies well within the realm of logic.
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| « Last Edit: Jun 30th, 2013, 11:28pm by 0.999... » |
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Mickey1
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Re: A real circus
« Reply #13 on: Jul 1st, 2013, 11:07am » |
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You are giving a similar example as on Wikipedia, “To give an informal example, for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection”, This amount to choosing the number 0,0000… in base 2. You can similarly choose any number in a countable set. It only reminds us that we can leave these sets behind us in our investigation of the real numbers.
The rest, the numbers which are not possible to define in a finite number of steps can only be referenced as a very amorphous group. Let us look at these numbers together, They can only be referenced as a group “those number which – if they existed - cannot be referenced”.
Therefore to “merely guarantee the existence of just one choice function for a family of sets” is exactly to postulate that this group exists.
“Postulation “ (by choice a function ) of real numbers actually is a better word than “definition”. The axiom concept assumes erroneously that they exist. Otherwise, the definition process would automatically guarantee that the choice function exists.
You can easily make this test:
1 Try to define (all) real numbers.
2 Test whether a choice function is hidden in the definition.
If you can’t find it I promise to help you.
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0.999...
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Re: A real circus
« Reply #14 on: Jul 1st, 2013, 6:41pm » |
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Quote:The rest, the numbers which are not possible to define in a finite number of steps can only be referenced as a very amorphous group. Let us look at these numbers together, They can only be referenced as a group “those number which – if they existed - cannot be referenced”.
Therefore to “merely guarantee the existence of just one choice function for a family of sets” is exactly to postulate that this group exists. |
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It is the complement of the set of all real numbers which can be constructed via a finite process. This set exists without appeal to the axiom of choice.
Quote:You can easily make this test:
1 Try to define (all) real numbers.
2 Test whether a choice function is hidden in the definition.
If you can’t find it I promise to help you.
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Proving that any such attempt involves (a weak form of) the axiom of choice is well within the realm of logic. Nevertheless, you end up being able to well-order the reals if you provide me with such a list of definitions, so I agree that it involves a weak form of the axiom of choice.
However, you are talking about the converse of what I think your point is. You actually have not provided such a construction of each real number in finitely many steps (involving the axiom of choice or not). It is impossible, of course, to do that without investing in infinitely many logical symbols, which would defeat the purpose obviously.
Finally, with the axiom of choice you can define via finite processes at most countably many individual real numbers in addition to what you had without using the axiom of choice. For instance, with a bit of forcing, and a hypothesis that certain extra axioms are consistent with ZF, one can show that non-measurable sets of real numbers do not have to exist; while in ZFC it is easy enough to construct one such set. Therefore, well-ordering the collection of all nonmeasurable sets in some interval of real numbers is sufficient to give a definition of a real number using the axiom of choice which is invalid without the axiom of choice.
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| « Last Edit: Jul 1st, 2013, 7:23pm by 0.999... » |
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Mickey1
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Re: A real circus
« Reply #15 on: Jul 2nd, 2013, 3:50pm » |
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I am not sure what a weak form of the axiom is although the Wikipedia article mentions several versions. New clickable terms, in need of additional clicks are included at a high rate.
For all my conviction, I am still at the mercy of Wikipedia, being an amateur. They mention a version of the axiom (the only one without too much theoretical jargon) in uncertain ways: “Informally put, the axiom of choice says that given any collection of bins…”. What “Informally put” means I don’t know, i) “similar to” ii) “true from a certain perspective but not all perspectives”, or perhaps iii) “not really true but it may it give you an idea”.
I get the impression that historically, the numbers in the number-line were the (viewed as unproblematic) real numbers and that running into the axiom was like an engineer running into practical problems which essentially have a philosophical root. I view the axiom of choice as a mathematical “engineering discovery” necessary to negotiate an obstacle along the road.
I can offer the notion from my own background the idea of model validation. Physicists have gone for a decade in circles before they understood that a model cannot really be validated since reality is not a model. Se also http://goneri.nuc.berkeley.edu/pages2009/slides/Jensen_Comments%20to%20t he%20students.pdf in which I comment (after the meeting) on some elements in the presentation of an earlier speaker, Allison Macfarlane (now chairperson of the US Nuclear Regulatory Commission).
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Mickey1
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Re: A real circus
« Reply #16 on: Jul 7th, 2013, 2:53pm » |
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A late addition:
In his Arxiv article “Pointwise Definable..” from 2012 Hamkins’ assumes the axiom of choice (“ZFC” is mentioned 5 times in the abstract).
“Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin”.
Therefore the AC implies a definition of the real numbers by instantaneous bookkeeping of an infinite number sequence such as all reals between 0 and 1. From this it would follow that while the ZFC falsifies the indefinability of reals, the ZF might not.
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Mickey1
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Re: A real circus
« Reply #17 on: Jul 7th, 2013, 2:53pm » |
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A late addition:
In his Arxiv article “Pointwise Definable..” from 2012 Hamkins’ assumes the axiom of choice (“ZFC” is mentioned 5 times in the abstract).
“Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin”.
Therefore the AC implies a definition of the real numbers by instantaneous bookkeeping of an infinite number sequence such as all reals between 0 and 1. From this it would follow that while the ZFC falsifies the indefinability of reals, the ZF might not.
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