[sritchie]

It follows from the way addition is defined on top of set theory. "a + b" is implemented as "increment a (the set that represents a) b times".

A number is represented in set theory as a set that contains all of the numbers before it. 0, 1, 2 is {}, {{}}, {{} {{}}}...

SO! If you start with a finite "a" and increment it infinite times, you still have infinity; you haven't broken out.

But if you start with Infinity, then adding anything to it gives you {Infinity}, {Infinity {Infinity}}, etc...

Transfinite addition is not commutative!

[RHSeeger]

Is addition defined _by_ set theory, or is set theory one way of defining addition? If it's the later, then there could be other ways of defining addition that don't have the same results for infinity (because our math system doesn't really "work" for infinity, or 0, depending on the circumstances).

I am in no way a mathematician. My question about the definition of addition as it relates to set theory is just that; a question.

[sritchie]

It's the latter; I'm also not a mathematician, just a guy who worked through Halmos's "Naive Set Theory" in intense detail...

But your question actually hints at my most profound takeaway from that whole book. I think what you're saying is right, AND that foundations-of-mathematics folks spent a long intense period searching for different set theory axioms that did NOT lead to transfinite numbers. But anything anyone could come up with that included "the axiom of infinity" led to transfinites leaking in.

Which begs the question of how to think about these things. Are they "real"? Are they an oddball side effect that we shouldn't take seriously?

I think you've arrowed right to the philosophical heart of all of this.

[chevman]

Does everything become a paradox given enough time and/or thought?

I think we often end up at the end of logical thought processes back at the original question - how can we observe and describe a system that we are inherently a part of?

[gowld]

There are many ways of definiting everything. Most of them are equivalent in the ways that matter, which is why math "works" so well as the language of science. Some of them are different in critical ways, which opens up vistas of new objects and concepts.

[bigdict]

I majored in math and my biggest problem with this is that you don't get to "do" anything infinitely many times in the math that I'm used to. In discrete contexts where infinity is used, you instead can "do" something an unbounded but finite number of times. In a continuous setting you are allowed to pick an arbitrarily large (finite) number.

In that context the first quantity that you refer to above is nonsensical because you can't "increment infinitely many times".

Secondly, I'm not sure your construction is correct, since your Infinity+1 set cannot be a singleton (it must contain all the numbers less than Infinity).

[sritchie]

Sorry, of course you're right on "Secondly". The right construction is ω, ω∪{ω}, ω∪{ω}∪{ω∪{ω}}...

For the first point, I went through the book long enough ago that I can't rebuild the proof here, but iirc the more rigorous idea is that you can construct a bijection between 1+ω and ω given the recipe I had above for how to represent numbers as sets, but you can't do it for ω+1, which is bijective with ω∪{ω}. The axiom of infinity declares that ω itself is a set, opening the door for transfinite numbers.

Better?

[bigdict]

Thanks, sorry for being pedantic. These sorts of constructions tend to trigger some kind of defense mechanism in me.

[sritchie]

No, of course you’re right to be! I owe myself another lap through this material and this is a good push…