Deeper than primes - Continuation 1/3*9

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Originally Posted by jsfisher (Post 13132004) |A| <= |B| if and only if / is defined by / means there exists an injection from A to B

You wrote

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Originally Posted by jsfisher (Post 13131810) Given some set A and some set B, |A| <= |B| is a proposition that may or may not be true

The focus is only on |A| <= |B| proposition, which can be written also as (|A| < |B|) OR not(|A| < |B|) (a tautology), exactly because not(|A| < |B|) can't be but (|A| = |B|) in case of |A| <= |B| proposition.

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Originally Posted by doronshadmi (Post 13131954) According to you, these properties can't establish the ZF(C) Axiom of infinity to actually be the ZF(C) Axiom of infinity unless more ZF(C) axiom are involved. Quote: Originally Posted by jsfisher (Post 13131810) Nope, I never said that. (And don't equate the name give to an axiom with what the axiom actually says. The Axiom of Infinity postulates the existence of a set with two properties. It does not call it an infinite set. Even if it had, that would not define what infinite set meant, just postulate the existence of one example.)

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Originally Posted by jsfisher (Post 13131810) The Axiom must to be coupled with other axioms to conclude von Neumann's ordinal is a set in ZF.

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Originally Posted by jsfisher (Post 13131810) Nope. You just need something that defines what "infinite set" means. "A set Q is infinite if and only if...."

Cardinality is a measure of the number of members of set A (notated as |A|)

Set A is called finite iff given any n in N, |A| is any particular n

Set A is called non-finite iff given any n in N, |A| is not any particular n

By the standard notion "given any" is the same as "for all" ( as seen in https://en.wikipedia.org/wiki/Universal_quantification ) but not in my framework, where "give any" holds for both finite and non-finite sets, where "for all" holds only for finite sets.

Non-finite sets have immediate or non-immediate successors exactly because given any n in N, |A| is not any particular n.

This is not the case with finite sets, they do not have immediate or non-immediate successors exactly because given any n in N, |A| is any particular n.

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Posted By:zooterkin

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