But we have to distinguish between the extensional concept of a number of hotel rooms that can be built, visited, observed, realized etc, versus the intensional concept of a countably infinite set of rooms. — sime
a piece of syntax representing an inductive definition of the natural numbers. — sime
There's no consideration of intensionality in the illustration. — TonesInDeepFreeze
extensionally meaningful — sime
So if one writes down an inductive definition of the natural numbers
1 + N <--> N
where <--> is defined to be an isomorphism — sime
then to say N is "Dedekind-Infinite" means nothing more than to restate that definition. — sime
Definition of 'extensionally meaningful? — TonesInDeepFreeze
That's not a definition of anything, let alone the set of natural numbers. — TonesInDeepFreeze
'N is Dedekind infinite' means that there is a 1-1 correspondence between N and a proper subset of N. There's no need to drag isomorphism into it. — TonesInDeepFreeze
'
The function {<j j+1> | j in N} is provably a 1-1 correspondence between N and a proper subset of N, so it proves that N is Dedekind infinite (and notice, contrary to your incorrect claim, choice is not involved). But that proof is not a definition of anything, let alone of the set of natural numbers. — TonesInDeepFreeze
extensional meaning — sime
the items it refers to — sime
definition of the set in terms of a formula — sime
category — sime
the presence of Choice causes all infinite sets in ZF to become Dedekind infinite by default — sime
which is a major failing of ZFC in ruling out the only sort of "infinite" sets that have any pretence of physical realisability. — sime
Undefinable real numbers have no place in my view.
— keystone
You reject vagueness then. That is certainly the usual thing to do. — apokrisis
And how do you know there is a rule unless you have ever seen some exception? — apokrisis
That is a better analogy. I prefer my own still – the static on the TV screen which is both every show you could ever see, but all at once ... or else just meaningless noise. — apokrisis
There's an Australian mathematician, Norman Wildberger, on YouTube who doesn't accept infinities.If you are experienced and trained in this area and would be up for helping me out through paid mentoring, please let me know. — keystone
Must an ordered field necessarily be a field of numbers?
— keystone
No. But all complete ordered fields are isomorphic with one another. So all complete ordered fields are isomorphic with the system of reals. — TonesInDeepFreeze
the axioms of set theory are not in concordance with the intuitive notions of 'finite sets'
— keystone
All the axioms are in that concordance, except one. — TonesInDeepFreeze
Hilbert's idea was that we can work in infinitistic mathematics if we have a finitistic proof of the consistency of infinitistic mathematics. Famously, we found out that there is no finitistic proof of the kinds of systems we'd like to use, not only not of set theory but even of PRA, the system itself that we may take as exemplifying finitistic reasoning at its "safest". Yet, if I understand correctly, Hilbert's condition was a sufficient condition not a necessary one. — TonesInDeepFreeze
That's interesting. But, if that is to be a statement in the system, we'd need to see "described" couched mathematically. I have a hunch that your notion is pretty much the same as 'there exist potentially infinite sets', and as I've said, I don't know a system that says it. — TonesInDeepFreeze
And yet Universal Turing Computation is a mathematical object – conceived in Platonia. This is the kind of "paradox" we are meant to be figuring out here, not simply saying one is the other as if the differences were moot. — apokrisis
Mathematics, in many branches, is brimming with sets. Analysis, topology, abstract algebra, probability, game theory... Can't even talk about them, can't get past page 10 in a textbook, without sets.
But of course, one can use the theorems of mathematics for engineering without tracing the proof of those theorems back to axioms, in particular the set theoretic axioms. That's not at issue. — TonesInDeepFreeze
A sequence is a set. And it has a domain, which is a set, and a range, which is a set. An infinite sequence is an infinite set with an infinite domain.
Of course, one can leave that unconsidered, not in mind, when working in certain parts of calculus. That is not at issue. But when we trace the proofs of the theorems of analysis back to axioms, then, in ordinary treatments, those are the axioms of set theory. — TonesInDeepFreeze
A circle is a certain kind of set of points. I don't know what you mean. — TonesInDeepFreeze
I am saying much the same thing. But the question is not where the numbers need to be represented or stored. It is how many decimal places do you really need for the task in hand? — apokrisis
I believe that calculus is more closely aligned with this parts-from-whole approach than it is with the conventional whole-from-parts approach. — keystone
(I had no way of knowing that, out of the blue, you would be using category theory) — TonesInDeepFreeze
But one can't fairly criticize the road of set theory if one is not addressing it as an axiomatization. And even if not criticizing set theory but instead just saying mathematics can be done with unformalized "potential infinity" instead, then it's not a fair comparison since one is an axiomatization and the other is not. — TonesInDeepFreeze
But when I started grad school at another university in 1962 one of the first required courses was an introduction to foundations using Halmos' Naive Set Theory and the Peano Axioms. It was quite illuminating. — jgill
For me, the problem is not so much that there is anything counter-intuitive about this, but rather that it's rude and bad business practice to keep waking guests up in the middle of the night and make them pack and move to another room, especially an infinite number of times. Not only that, but the poster keystone has added lamps that keep turning off and on, which is extremely annoying when people are trying to get a good night's rest for the next day when everybody is going out to see Zeno's 10K Charity Run where Achilles will have to run through an infinite number of distances and suffer the ignominy of getting beat by a turtle. — TonesInDeepFreeze
the very fact that anytime you have probability with infinitely many "contestants", whether it's dense space or whatever, you will necessarily either give the "contestants" a probability of 0 or be faced with adding up over 100% (since reiteratively summing any non-zero quantity indefinitely will approach over 100% at some point).
Your "solution" isn't a solution in that it doesn't talk about what the problem talks about. The "problem" is referring to continuity in dense contexts: it's not at all a "problem" in nondense contexts, this is equivalent to solving the Liar paradox by just saying "what if the guy doesn't lie?" — Kuro
I believe that cuts made to a continuum are perfectly precise since I can draw it with no vagueness. For example, consider this drawing of y=0 and y=x^2-2:
There's no blurriness to my drawing. However, when I start to measure it (usually through calculus), my measurements may be imprecise. — keystone
Your analogy betwen mathematics and theology is not apt.
One can disprove 'there exists an infinite set' by stating axioms that disprove 'there exists and infinite set'. The obvious choice for such an axiom is 'there does not exist an infinite set'.
Anyway, I never asked you to disprove anything at all. — TonesInDeepFreeze
We don't intend or claim that a domain of discourse for set theory is a world such as a physical world of physical particles and physical objects. At the beginning of this discussion, if asked, I would concede that immediately. — TonesInDeepFreeze
You've described your notion of potential infinity a few times (in another thread especially). And I've replied about it each time. Now, you're coming back to restate it, but still not addressing the substance of my previous replies. As in another thread, this just brings us around full circle. — TonesInDeepFreeze
The thought experiment is suggestive of an analogy with set theory, but suggestiveness is not an argument about set theory itself. — TonesInDeepFreeze
why would we think they can be completed in reality?
— keystone
We don't! Set theory doesn't say there's a "completion in reality". Set theory doesn't have that vocabulary. — TonesInDeepFreeze
There's an Australian mathematician, Norman Wildberger, on YouTube who doesn't accept infinities.
Here's a link to one of his videos.
Difficulties with real numbers as infinite decimals
https://www.youtube.com/watch?v=tXhtYsljEvY
You might try contacting him. — Art48
How would you catalogue all continuous curves? That would be a starting "point". In order to have derivatives and integrals you would need some kind of function derivable from a catalogued example. Sorry, but the whole approach sounds absurd. — jgill
This is Russell's argument. — apokrisis
But the reverse argument also applies. The representation can be sharper than what it represents. The right facial recognition algorithm could separate a dim CCTV image of Keystone in a hoody from all the other faces stored in a police data bank. Signal processing can extract structural information that stands behind any amount of confusing surface detail. — apokrisis
But how wide are your lines – even mathematically? — apokrisis
How sure are you they are single lines and not a small bundle of lines sharing a neighbourhood with infinitesimal spacing? And when does this vagueness start to matter? — apokrisis
Doesn't it matter if your rigorous mathematical edifice must also fit a physical world were nonlinearity is in fact the generic condition? — apokrisis
But that doesn't engage with the foundational issue of whether reality itself is vague or crisp at base. And hence what kind of ontology we are correct to import into our "picturing" of math's epistemology. — apokrisis
But it turns out his "fair case" is just to wave his magic wand (actually he uses an old drum stick) and say that the natural numbers are the foundation for all the other branches of mathematics. But he says not a single word showing how that would be done except for a chart with 'the natural numbers' as the base of the pyramid of mathematical subjects. — TonesInDeepFreeze
if we've only proved that the reals are an ordered field, then is it possible that we haven't proved that sqrt(2) is a number? — keystone
When I think of A being equinumerous to B, I think that there exists a bijection AND no injection between A and B.
When I think of A being more numerous than B, I think that there exists an injection from B to A AND none from A to B.
When I think of A being less numerous than B, I think that there exists an injections from A to B AND none from B to A. — keystone
my intuition based on finite sets leads me to believe that infinite sets are all empty. — keystone
I'm not saying that set theory is wrong, I'm just proposing that set theory might not be about actually infinite sets, but instead the potentially infinite algorithms that describe the infinite sets. — keystone
Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points? — keystone
Can you explain what you mean by 'catalogue all continuous curves'? — keystone
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