> Woodin named the axiom (*), pronounced “star,” because it was “like a bright source — a source of structure, a source of light,” he told me.
There are only two hard problems in mathematics: infinity and naming things.
How many real numbers exist? New proof moves closer to an answer
151–160 of 359 posts
Re: How many real numbers exist? New proof moves closer to an answer
#152Earlier quoted context omitted.
> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…
why is that "unsound"? what's wrong with an unwritable natural? Almost all reals are unwritable.
I wonder what you mean by "what's wrong with an unwritable natural"... It is sound to assume one! And it doesn't make any true mathematical facts false. But of course it's sound to assume many silly things. But why assume those things?
The reality is that mathematicians didn't first come up with the Peano axioms and then study the interesting consequences of them. Both as a matter of history and also as a matter of why most mathematicians do math, mathematicians first came up with numbers, and only later came up with axioms that allow them to do rigorous reasoning about them. The same is actually true of almost all math—real numbers were invented to do rigorous reasoning about calculus, which by that point had been around for a century plus; set theory likewise.
In other words, if the goal is to come up with any old consistent set of axioms, you can assume an unwritable natural number, that's fine, go ahead. But if your goal is to study natural numbers—which, like, I have a lot of experience with natural numbers in my day to day life, I'm pretty sure I could write all of them if I had enough time and space and so on—then you want to study natural numbers, not some other weird things where there are unwritable weird things.
Re: How many real numbers exist? New proof moves closer to an answer
#153Maybe I misunderstood the article but if the set of real numbers is finite then it should be countable. But I can easily prove that the set of real numbers or any subset of real numbers is not countable. Been a really long time since I’ve thought about this but wondering what I’m missing.
How can you easily prove that the set of real numbers is not countable? I don't think it's as easy as you claim, but I'm kind of a dummy so it's quite probably I'm wrong.
Here's some background on the proof [1]. Here's a video explaining it little better [2].
[1] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument [2] https://www.youtube.com/watch?v=elvOZm0d4H0
Re: How many real numbers exist? New proof moves closer to an answer
#154Earlier quoted context omitted.
Thanks for the reply. Question though. in Cantor's argument we explicitly mapped the reals to aleph-0 so it makes sense that our conclusion decides that mapping to aleph-0 is too small so it's size must be larger. Where in the forcing process do we even "use" aleph-1? If we used aleph-1 then it could see the parallels and the argument would make sense - but all I see in the forcing process is "start with a set of all…
Great question, I have no answer. The article explained forcing in such a way as to simply restate what I thought we already knew: given a real, there is no "next" real. (ie, there are a non-countable-infinite number of reals between any two reals). I don't see the newness that forcing brings to this.
Forcing is about a completely different question.
The question is not where there are reals between reals, it's a question of the size of sets. In particular is there a set strictly larger than the natural numbers, but strictly smaller than the reals? Forcing allows us to construct such a set in ZFC.
Re: How many real numbers exist? New proof moves closer to an answer
#155Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists. In other words the article a…
Individually, or independently, axioms have no truth value. But when you put together a system of multiple axioms, they can contradict eachother. I see no problem with the question of whether the continuum hypothesis is true, given ZFC as a precondition. All we are asking is if the axiom contradicts ZFC. Axiom independence is very similar to operator commutation in quantum mechanics.
Re: How many real numbers exist? New proof moves closer to an answer
#156Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists. In other words the article a…
Individually, or independently, axioms have no truth value. But when you put together a system of multiple axioms, they can contradict eachother. I see no problem with the question of whether the continuum hypothesis is true, given ZFC as a precondition. All we are asking is if the axiom contradicts ZFC. Axiom independence is very similar to operator commutation in quantum mechanics.
The Continuum Hypothesis and its negation are both proven to be consistent with ZFC (this is what the article is talking about RE forcing and Godel's proof of the consistency of CH). There is no contradiction to assume one or the other alongside ZFC. The article is really talking about Platonic truth when it talks about something being true or false.
> Individually, or independently, axioms have no truth value.
Indeed, you are not a Platonist :).
Re: How many real numbers exist? New proof moves closer to an answer
#157As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist. What is interesting is that as shocking as constructivism may be, there is no logical flaw in it…
> there are only a countable number of real numbers Then you should be able to come up with a function that assigns a natural number uniquely to each real number. Of course if you tried that I could immediately name you a real number, or a pair of them, for which your rule doesn't work.
That depends on what you mean by "assigns uniquely", "rule" and "doesn't work", which is why this question is deeply entangled with philosophical issues that cannot be settled purely mathematically.
It is obvious that all expressions in the English language can be ordered from smallest to largest and lexicographically, which makes these expressions trivially countable. We can thus assign natural numbers to real numbers by assigning numbers to their expressions in a natural or formal language, which will of course include infinitely many expressions that are just nonsense descriptions and infinitely many expressions that map to the same real number. These expressions will also include any possible expressions of Cantor's or other diagonalized numbers. In such a sense then, we can trivially "count" the real numbers unless we hold the philosophical view that there are real numbers that are not expressible. This is where it becomes a question of philosophy of mathematics, not mathematics proper.
You can of course object that what you meant by "assigns uniquely" is an unambiguous 1:1 mapping and that including any number of nonsense descriptions misses the point. In that case giving a "rule doesn't work" because the diagonalized number always escapes the proposed system of counting the numbers, but only because the diagonalized number is allowed to 'parasitically' depend on the totality of the system, but is excluded from the system (or else it would diagonalize itself and become ambiguous at that particular decimal place). This particular viewpoint is tied to a particular philosophical position, however, and not all positions in the philosophy of mathematics will agree with it.
This all might seem trivial or even nonsensical (as philosophy of mathematics so often appears), but I merely want to point out that the 'uncountability' of the real numbers is not a consequence of the set of the natural numbers being 'too small' to hold all the real numbers, because they are 'large enough' to assign numbers to all possible descriptions all real numbers that will ever be expressed in language. Uncountability is a consequence of a view that restricts Cantor's diagonalized number from the set of the countable number but still considers this diagonalized number to be a real number (which again is only unambiguously defined if it is not allowed to diagonalize itself). There are however other possible philosophical viewpoints which either include the diagonalized number in the set of countable numbers (at the cost of including ambiguous or paradoxical numbers) or reject the view that Cantor's diagonalized number should be considered to be a real number in the first place.
tl;dr: Yeah, you can always name a real number for which a particular counting rule does not work, but only as long as there is agreement regarding the philosophical underpinnings. Most mathematicians can probably be considered platonists and from their standpoint the real numbers are obviously uncountable, but that is by no means true for all positions in the philosophy of mathematics.
Re: How many real numbers exist? New proof moves closer to an answer
#158Maybe I misunderstood the article but if the set of real numbers is finite then it should be countable. But I can easily prove that the set of real numbers or any subset of real numbers is not countable. Been a really long time since I’ve thought about this but wondering what I’m missing.
No, what the article is talking about is the question whether or not the cardinality of real numbers is the smallest uncountable infinity or some other, larger uncountable infinity. The only countable infinity is aleph-0, the cardinality of natural numbers, and Cantor showed that aleph-0 is too small to hold all reals. So reals must be uncountable, but there is an infinite hierarchy of uncountable infinities, and it…
unrelated: how do we know there are no alephs between 0 and 1?
Re: How many real numbers exist? New proof moves closer to an answer
#159The only thing this proves is that mathematics is a soft science, where concepts like "number" and "infinite" are subjective. There are obviously infinite numbers, if you think there's a finite number of numbers, take that number and add one to that. QED
" Please don't post shallow dismissals, especially of other people's work. A good critical comment teaches us something. " https://news.ycombinator.com/newsguidelines.html
I can't believe you called this shallow and didn't even include an explanation as to why, ironic. There's nothing shallow about this at all, it goes to the heart of this fake intellectualism on this site.
Re: How many real numbers exist? New proof moves closer to an answer
#160Earlier quoted context omitted.
"Didn't you just conclude that it's impossible to have a set of all real numbers?" Cantor's diagonalization proof proves that it's impossible to list all the real numbers with a list of size aleph-0, which is the cardinality of the set of natural numbers. The forcing proof is an attempt to prove you also can't do it with the a list of the size aleph-1, which is the size of the power set of aleph-0. It purports to pro…
That's weird. I thought that it was proven that existence of sets larger than aleph-0 but smaller than the number of real numbers is undecidable and you can add it (or negation of it) as additional axiom to math. https://en.wikipedia.org/wiki/Cardinality_of_the_continuum "The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between aleph-0 and c=aleph-1. The truth or falsity of…
This is covered in the article.