Re: An uncountable countable set

Tony Orlow wrote:
Also, upon which axioms is the definition of cardinality based?

The usual definition is:

card(x) = the least ordinal equinumerous with x

The definition ultimately reverts to the 1-place predicate symbol 'e'
(and the 1-place predicate symbol '=', if equality is taken as
primitive). For the definition to "work out" ('work out' is informal
here) in Z set theory, we usually suppose the axioims of Z set theory
plus the axiom of schema of replacement (thus we're in ZF) and the
axiom of choice (thus we're in ZFC). However, there is a way to avoid
the axiom of choice by using the axiom of regularity instead with a
somewhat different definition from just 'least ordinal equinumerous
with'. Also, we could adopt a "midpoint" between the axiom schema of
replacement and the axiom of choice by adopting the numeration theorem
(AxEy y is an ordinal equinumerous with x) instead, which would be a
method stronger than adopting the axiom of choice, but weaker than
adopting both the axiom of choice and the axiom schema of replacement.
As to the more basic axioms of Z, for the definition to "work out", I'm
pretty sure we need extensionality, schema of separation (or schema of
replacement if we go that way), union, and pairing (pairing is not
needed if we have the schema of replacement). I'm not 100% sure, but my
strong guess is that we don't need the power set axiom for this
purpose. And we don't need the axiom of infinity.

Why don't you just a set theory textbook?

MoeBlee

.

Relevant Pages

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  • Re: An uncountable countable set
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  • Re: My last theory.
    ... Let us see how this Replacement schema works. ... in Axiom Schema 8. ... Schema only proves the existence of the images of functions -- ...
    (sci.logic)
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