Do we have ways of computing orders of infinity higher than ℵ₀?
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barsoap 10 points 3 years ago
Symbolically, sure, but then you're not dealing with infinities you're just representing them.
It's a meme it's playing fast and loose with things but the general gist is that mathematics, to this day, doesn't really care about Gödel/Church/Turing, incompleteness, the halting problem, whatever angle you want to look at it from. Formalists lost the war and they simply went on doing maths as if nothing had happened, as if a system could be simultaneously complete and consistent. There's people out there preaching to the unenlightened masses but it's an uphill battle.
kogasa 6 points 3 years ago
Math went on because it doesn't matter. Nobody cares about incompleteness. If you can prove ZFC is inconsistent, do it and we'll all move to a new system and most of us wouldn't even notice (since nobody references the axioms outside of set theorists and logicians anyway). If you can prove it's incomplete, do it and nobody will care since the culprit will be an arcane theorem far outside the realm of non-logic fields of math.
barsoap 2 points 3 years ago
You wouldn't even notice if some proof is wrong because it relies on an inconsistency that's the issue. And that's before you didn't notice because noone builds anything on axioms but instead uses fragile foundations made of intuition, hand-waving, and mass psychology.
Incomplete is fine that's exactly what constructive maths is doing.
kogasa 1 point 3 years ago
You wouldn’t even notice if some proof is wrong because it relies on an inconsistency that’s the issue.
You wouldn't notice because there's no realistic chance that any meaningful result in the vast majority of math depends strictly on the particular way in which ZFC is hypothetically inconsistent.
And that’s before you didn’t notice because noone builds anything on axioms but instead uses fragile foundations made of intuition, hand-waving, and mass psychology.
This is a ridiculous attitude. Nobody uses the axioms of ZFC directly because that would be stupid. It's obviously sufficient to know how to do so. There is literally no difference to the vast majority of all math which particular axiomatic formalism you decide to use, because all of those results are trivially translatable between them.
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uriel238 2 points 3 years ago
We have sorta the same problem with imaginary numbers, and I remember some programmable calculators can process complex numbers using symbolic representation (which happens to work similarly to Cartesian coordinates, so that's convenient)
But from what I remember any infinity bigger than counting numbers (say the set of real numbers) cannot be differentiated from each other, so we don't have established rules.
To be fair, I last tinkered with infinities in the aughts and then as a hobbyist. The Grand Hilbert Hotel can accomodate more compound infinities and still retain perfect utilization since the last time I visited.
barsoap 3 points 3 years ago
https://en.wikipedia.org/wiki/Continuum_hypothesis
Hmm. Frankly speaking I always assumed Mathematicians had more of an idea about infinities I mean why even have indices if you don't have an inductive rule to descr... oh wait never mind.
That said the reals aren't countable yet we have perfectly reasonable ways to deal with them symbolically, even compute with them, represent every single one of^1^ them in finite space, it's just when you want to compare or output them with infinite precision that you might have to wait for eternity. But who needs infinite precision anyway, arbitrary precision is plenty.
^1^ On second thought, after diagonalisation, no we don't. Or we do because there's some magic going on with included transcendental constants that break through that do I look like a numerologist.
kogasa 2 points 3 years ago
I don't understand what you think the problem is. What do you mean infinities can't be differentiated from each other? Infinite cardinals are by definition equivalence classes of sets that can be placed in bijection with one another. You can compare one cardinal to another by showing there exists an injection from a representative set of the first one into a representative for the other. You can show equality by showing there is an injection both ways (Cantor-Schroder-Bernstein theorem) or otherwise producing a bijection explicitly. Infinite ordinals may as well be copies of the natural numbers indexed by infinite cardinals, so of course we can distinguish them too.
uriel238 1 point 3 years ago
So far AFAIK we have two kinds of infinity: Those that can be accommodated at the Grand Hilbert (e.g. integers, fractions, etc.) and those that cannot (set of irrational numbers, set of curves, set of polytopes, etc.) This was why we had to differentiate orders of infinity, e.g. ℵ₀ (The Grand Hilbert set), ℵ₁ (the irrational set, the real set), ℵ₂ (???), ℵ₃ (?????), ℵₙ (??!??????????)
For values of infinity that are in higher orders than ℵ₀, we can only tell if they're equal to ℵ₁ or undetermined, which means their infinity size is ℵ₁ or greater, but still unknown.
Unless someone did some Nobel prize worthy work in mathematics that I haven't heard about which is quite possible.
spacesweedkid27 14 points 3 years ago
Hello I am computer scientist and I can tell you that nobody of us would say something like that to the mathematician. Very very very many concepts of Computer Science are based on mathematics that require the concept of infinity to work like for example derivatives.
niktemadur 5 points 3 years ago
[TREE(TREE)] ↑↑↑ (Busy Beaver)
SendMeBakedBeans 4 points 3 years ago
This is just the other side of the "We're well adapted to this world so it must have been made for us" coin.
uriel238 2 points 3 years ago
That gets into the invention vs. discovery dilemma, but mathematics is just a rule system with which we can make models of natural phenomena. Despite multiple efforts to try to resolve all paradoxes, we instead eventually developed a proof that no sufficiently complex system of mathematics can be made without paradox.
AnUnusualRelic 8 points 3 years ago
So that they can be taxed, of course.
kogasa 2 points 3 years ago
Computable. There are countably many computable numbers since there are only countably many possible programs. Non-computable numbers can't be exactly referred to / described / constructed by a program, so if your point of view is that everything is a program, you would say they don't exist.
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Starter mathematicians might think everything is math.
Senior mathematicians lose their sleep whenever they think about what our math is missing and they don't know what it is yet
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