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07:04
@Wildcard As I said, the diagonal argument does not depend on you assuming that standard real numbers exist. It does show that there is no computable list of all computable reals. A computable real is just a certain kind of program, and so there is completely no "infinite" thing in this claim. Literally, a program exists that, when given as input a enumerator of computable reals, will give as output a computable real that it does not enumerate. This is shown by the diagonal argument.
You can validly object that "there is no (countable) list of all standard reals" does not make sense if you already reject the standard construction of reals. However, you cannot validly object to "there is no computable list of all computable reals", unless you object to meaningfulness of programs.
Anyway, my point stands that you misunderstood the diagonal argument. It did not assume "you can have an arbitrary infinite sequence of digits which cannot even be described". It shows that if you do have a countable list of such objects, then you can construct a new one that is not on that list. If you object to the manner of construction as not producing an actual object, then the argument fails for standard reals but still works for computable reals.
Does what I say make sense now?
07:33
@user21820 moreso.
@user21820 to be honest, this argument doesn't hold from the fundamental axiom I stated earlier.
The "real number" that is to be computed which is to be not on my list, can never be computed all the way out to the end. It would have to be so computed in order to be proven not to be on the list. More precisely, the end result of the computation cannot be referred to as a real number in the formulation I am taking here. Only the numbers computed at any arbitrary point along the way can be referred to.
@user21820 there are several negatives in this comment which made it hard to parse, but I think I got it. And yes, I probably do reject the standard construction of reals.
In my own philosophy of mathematics (and again, I'm not trying to "convert" anyone), (a) there is no such thing as a non-computable number.
Therefore, all numbers are computable (or else they're not numbers), and thus the inclusion or omission of the word "computable" from the word "number" can't change the meaning of a statement, only make it more or less clearly communicative. :)
As for "real numbers"—I would define a real number (by contrast with a rational number) roughly as a computable process which yields rational numbers with successively greater precision.
Significant digits gives a good idea of what this looks like.
So the number "2" is a particular number. The number "2.000" is a more precise number. Any number is inseparable from the degree of precision of that number. So numbers have to be taken in context. They can have arbitrary precision, but never absolute precision.
Again, I shall emphasize (repeatedly) that this is my philosophy of mathematics.
All of high school mathematics functions perfectly under this interpretation. It's only when you get to imponderables such as infinite cardinalities and unspecifiable numbers that you would come a cropper.
So, back to computable reals and the diagonal argument.
I earlier (in a comment on Philosophy SE) asked you a question which bears repeating, as part of this discussion:
> Why can you not write a program which, given any program which enumerates a list of rational numbers, outputs a rational number not on the list?
I suppose I'll be happy to back off from my statement about computable reals, because I consider that a contradiction in terms in the first place in the way it is standardly interpreted.
Instead, what is typically called a "computable real number" I would instead call a computable sequence of ever more precise rational numbers.
You can never reach the end of the sequence. So the "real number" isn't really real at all. Just like "infinity" isn't a number, an infinite sequence of non-repeating digits isn't a number either.
(I'm enjoying this discussion a lot, by the way.) :)
Incidentally, it may have escaped your notice—it almost escaped mine; I had to reread my comments to verify—but I never made the statement that the real numbers are countable. What I said was:
> From this philosophical vantage, the "uncountability" of anything becomes impossible to prove.
@user21820 I understand the standard interpretation, actually. I don't even disagree with it as a conditional statement—if you accept the premises, you must accept the conclusion. From my own premises as stated, though, the diagonal argument becomes indistinguishable from an argument for the uncountability of natural numbers:
"No matter what numbers you put on the list, I can find a bigger number that's not on the list."
 
4 hours later…
user84215
11:55
in Mathematics, 18 mins ago, by MathematicsAminPhysics
The room Math Workshops has been created.
user84215
in Math Workshops, 7 hours ago, by MathematicsAminPhysics
This room has been created to hold math workshops. Please inform me of your suggestions to create events for them. Please note that your suggestions need not be very sophisticated; they can be elementary.
12:57
@Wildcard I don't know whether you realize that your position is inconsistent. The diagonal argument is totally valid even when your function-types and power-types are completely constructive. Namely, let N denote the type of naturals, and let S = Func(N,Bool) denote the type of (computable) functions from N to booleans, we can consider S to represent the type of programs that output binary strings, which can represent computable reals (but not uniquely).
Then the diagonal argument gives a proof that there is no (computable) function that surjects from N onto S.
I put "computable" in brackets because you said in your view every real is computable, which implies that function is computable. I'm saying that the diagonal argument is totally unaffected by your choice of view.
If there was such a surjection F, then let g = ( N k ↦ ¬F(k)(k) ) and so g ∈ Func(N,bool) but g ≢ F(k) for every k ∈ N. All this is constructive, since F(k) literally disagrees with g on input k.
This is for the function-type. So to apply to computable reals, you need the same patch to ensure non-zero difference. In the constructive setting we cannot always determine inequality between two (computable) reals but we sometimes can, and the standard patch does work constructively.
Therefore, I will repeat that if you do not agree that the diagonal argument shows that there does not exist a computable list of all computable reals, then you have to pinpoint a specific deduction step in the proof and a specific reason for rejecting it. Otherwise you are honestly just beating about the bush.
@Wildcard Note that I never said that you said it either.
@Wildcard This is false. I'm sure you can figure out why.
 
1 hour later…
14:11
Please note that MathematicsAminPhysics is a semi-crank. He posted in the logic chat-room for the first time here, essentially ignoring the serious mathematical discussion. He did not answer meaningfully to my responses but tried to twist my words. The second time was here and again he tried to twist my words.
The third time was here. He simply repeated his question for no rhyme or reason, as if the previous conversation had never occurred. This does not seem to be a serious inquiry even of the philosophical sort. I'm not the only one who thinks he has ulterior motives and is not serious about studying mathematics or logic. See this comment.
Another of his crank theories can be found here.
Note that he attempted to push his crank theory about the "human mind" thing multiple times in multiple places. See here to find some of them.
 
6 hours later…
20:28
@user21820 Actually, it is exactly and literally true. No matter to what arbitrary length you extend your list of natural numbers, there exist greater numbers which are not on your list. That is the exact definition of infinity: something which cannot be completed. There is no complete list of natural numbers. There is a means of computing any natural number. I don't think you can precisely define the word "countable" from these axioms, let alone "uncountable."
@user21820 I don't quite follow this notation. What is k?
@user21820 I reject the axiom "there is an infinite set." (Regardless of how you phrase it.) Therefore discussions of surjections, injections or bijections on the natural numbers are meaningless (totally undefined), because the natural numbers do not form a set.
You can define a set of natural numbers of arbitrary size, and you can define an algorithm (even just iteration of the successor function) which provably continues indefinitely to produce more numbers, but you cannot define a "set" of infinite cardinality. It's not a set anymore.
@user21820 That's ad hominem, incidentally. And no reason to attack someone. Looks to me like he just wants to have a philosophical discussion about math, and the philosophy of mathematics. No reason to get caustic about it.
20:53
@user21820 By the way, I agree he's not serious about studying mathematics or logic; he seems more interested in creating his own mathematics or logic (and one which would not generally be called mathematical or logical; just philosophical). But I don't see what that has to do with ulterior motives.
Obviously he's wildly out of agreement with the purpose of the Math Stack Exchange and MSE chat, so yeah, I'm not going to support totally changing the purpose of the site. :) I wish him the best of luck setting up what he wants elsewhere as his own site....

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