First, we realise if two different types of infinity existed, they would have to be larger than each other. Thats a logical contradiction, so we must of made a wrong assumption; only one kind of infinity can exist. — Devans99
The demonstration is quite easy (https://en.wikipedia.org/wiki/Cantor%27s_theorem). But there is a problem with the statement of the theorem: Russel's paradox (https://en.wikipedia.org/wiki/Russell%27s_paradox). The concept of "set of all subsets" is contradictory. — Mephist
So, in a sense, from the point of view of logic, all infinites are only "potential" — Mephist
If you use first order logics on the domain of real numbers, the set of all subsets of real numbers is the same thing as "the set of all sets" — Mephist
The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.
ZFC is an axiom system formulated in first-order logic ...
You have claimed that Russell's paradox invalidates the powerset axiom but I still don't follow your logic. In fact if the powerset axiom were false, I would have heard about it. — fishfry
What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica — Mephist
Every non contradictory axiomatic theory based on first order logic has a finite (non-standard) model — Mephist
Perhaps you're thinking of the kind of set theory used by Frege — fishfry
Oh my, no. Not at all. You should read the link you posted. There's no nonstandard finite model of ZF — fishfry
"the set B is bigger than the set A if there isn't any function that for every element of A gives an element of B and covers all B" ( i.e. each element of B corresponds to some element of A ) — Mephist
This is what it means "an infinite hierarchy of of infinite sets each one bigger than the other". — Mephist
In the standard contemporary mathematics based on set theory you can't speak of the "set of all sets" because it's not a set itself — Mephist
What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica — Mephist
This is a nonsensical definition: for instance, it claims the even numbers are the same size as the natural numbers (as there is a one-to-one correspondence between the two). But the even numbers are a proper subset of the natural numbers. If either had a size, the size of the natural numbers must be greater than the size of the even numbers. — Devans99
Clearly you know how many even numbers there are and you know how many natural numbers there are. And since you argue the set of natural number is the larger, then in consequence it must be possible to count the even numbers. Please do so and tell us how many even numbers there are. — tim wood
If you read any of the above, you would have gathered that I maintain infinity does not have a size so it is impossible to measure the size of infinite sets. — Devans99
infinity is not a number — Devans99
And then you have to be careful, which you neither are nor appreciate the need to be, with the concepts applied to transfinite sets. That is, basic arithmetic functions don't work quite the same way. And so on. — tim wood
This isn't news to mathematicians. When the concept of infinity was invented, there was a (perceived) need for it to be integrated into mathematics. (The alternative was to leave infinity standing alone and lonely, and this (apparently) was unacceptable.) The mess you observe is the result of that 'integration'. — Pattern-chaser
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