Infinity Explained By Leading Expert
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- Опубликовано: 8 фев 2025
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Joel David Hamkins is a Mathematician and Philosopher who researches infinity. He is the O'Hara Professor of Philosophy and Mathematics at the University of Notre Dame.
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I think he mostly only discussed two kinds of infinities: cardinalities and ordinals.
However, there’s also limits, hyperreals, and points-at-infinity. Sometimes mathematicians will just define a new infinity for a specific purpose, like when we need an element larger than everything in a set, we just adjoin an infinity symbol defined to be the largest.
Each of these infinities are different and have their own niche uses.
There’s also nothing wrong with such variety, since each one has its own rigorous definition for its own context. It doesn’t really make sense to have a single notion of infinity.
scroll to 5:38 in the video for the answers, everything before that is conjecture.
Take a set. Does there exist a bijection from a proper subset to your original set? If so, then your set is infinite.
In layman's terms, can you find a one-to-one correspondence between your set and a proper subset of it?
For example, if you see three sheep and three cows, you can put them both into a one-to-one correspondence with each other. You can even put the three sheep in a one-to-one correspondence among themselves. Each sheep corresponds to itself, or maybe the sheep to its right, etc. The one thing you can't do is put the three sheep into a one-to-one correspondence with the proper subset of only two of the sheep. You will always have one sheep left "un-corresponded". This is why we say 3 is finite. For infinite sets, you don't have this issue. I therefore think of this as the defining property of infinity.
The point about transfinite ordinals is more of a red herring, in my opinion. There are nuances of what it means to be an ordinal that are misunderstood. For example, pertaining to distinguishing successor ordinals and limit ordinals.
The first, smallest ordinal ω_0 is defined as the smallest non-empty inductive set. It corresponds to the smallest cardinal, ℵ_0. But even then, not all limit ordinals correspond to infinite cardinals, so it can be a point of confusion.
I would argue that when most people think of infinity, they are thinking of it in a cardinal sense, not ordinal sense. I would simultaneously argue that the ordinal sense can be more interesting, because it leads you to discovering the next largest cardinal, ℵ_1. Is that the same as the cardinality of the continuum? Boom! Continuum Hypothesis. Can you keep succeeding to bigger and bigger cardinals by finding bigger and bigger ordinals? Yes. But what if there were cardinals so big, they weren't in this ordering? Boom! Inaccessible cardinals.
I expected the expert to be bigger.
Classical mathematics is so boooring. Everything revolves around excluded middle.
Either(P, not-P)
A set is either finite or non-finite (e.g infinite)
A set is either infinite or non-infinite (e.g finite)
If excluded middle is seen as a false dichotomy then suddenly we get to talk about neither infinite nor finite objects...
If Omega can be added to, I think it's clear that it's not infinite, nor infinity.
Thanks for your comment. Interesting point. Do you think infinity exists at all then, or just that Omega can't equal infinity?
@@thehumanpodcastofficial Yes to both. I think infinity is the reality out of which finite ideas are abstracted by mind and senses; and that omega cannot equal it if we can add anything to it. Infinity is non-finite, so it can't be limited or defined. If you can add something, you've limited it, and so it isn't infinity, just another finite concept used to represent or symbolise an idea of infinity that is, necessarily, finite.
There are different kinds of infinity. The precise definitions mathematicians use show there is even variety in the same kind of infinity.
@@JM-us3fr I've seen this idea that there are different kinds of infinity. The different kinds of infinity are always different ways of counting. It may be a mathematical term used in a different way, but philosophically, how are we saying that infinity is constructible through simply adding finite values together?
Where do finite values come from? - our perception via the mind and senses, don't you think? So my third question to you now is, by declaring infinity to be a succession of finite intervals, are we saying that the way humans perceive the world is how the world actually is?
@@leosphilosophy If you want to better understand the philosophy of infinity, I would recommend Graham Oppy’s book _Philosophical Perspectives on Infinity_ . He covers a lot of the different kinds, and goes into whether they actually exist.
Personally, I’m not interested in a physically real infinity, just as I’m not interested in a physically real mathematics. Mathematics is about the logical consistency of concepts, and I don’t view concepts as existing. Perhaps there is a real-world analog of some of these concepts, but it doesn’t matter either way. It’s best that you don’t confuse the difference between _conception_ and _perception_ .
Also, the different infinities are not just different kinds of counting. For example, the real numbers can’t be counted; not because it never ends, but because there isn’t even a procedure in principle that would hit every real number. So the real numbers have a _larger_ infinity than countable infinity (in a precise way). Furthermore, infinite limit just means boundless, which has nothing to do with counting. Points at infinity are equivalence classes of lines, which again doesn’t have anything to do with counting.
Also, a “succession of finite intervals” is not really how anyone defines infinity. A single interval by itself is an infinite set (infinitely many points), and is often of finite length (max* difference). We have to be careful which definition we are using in which context.
After you remove the uhh's, and the ahh's, and the so-on and so-on from this video it will be -1 seconds long and still explain what infinity is.
You can *always* add one? What if you run out of 1s? You can *never* run out of 1s?
You could've just said you are engaging in viciously circular reasoning.
I guess that's axiomatics/Mathematics and infinite-time computations for you...
Actually you only need to add one once to show there is no largest number. Suppose N is the largest number. Then consider N+1. This shows N was not largest. I didn't do anything infinite there.
@@grumpyparsnip Suppose N is the largest number. By definition such a number would have no successor e.g succ(N) is undefined.
Then consider N+1 is a malformed expression.
This shows that you don't understand your supposition.
@@tgenov indeed, but succ(N) is always defined, thus the contradiction. It is true that you could set up an alternate axiom system in which succ(N) is not always defined, but the beauty is that infinity arises out of very simple axioms, whereas it is very awkward to try to formulate a logically consistent ultra-finitist system of numbers. You'd have to qualify equations like a=a+b-b because sometimes a+b is no longer a number, even if a is small enough. It would make for very tedious algebra.
@@grumpyparsnip “alternative” axiom system? To what?
You can suppose an axiom system with a largest number.
Or you can suppose an axiom system with a total successor function.
You can’t suppose both without disagreeing with yourself.
Chose one.
@@tgenov we agree on the fact that largest number and the existence of a successor are inconsistent. Alternative to the usual ZFC axioms, or even Peano arithmetic if you like.