Re: Relative Cardinality

If you get to talking about rationals and irrationals as being
basically interleaved wihin the real numbers, to do so the reals are a
contiguous sequence as well as complete ordered field, or as Tim called
them the Finlayson numbers, or pseudo-reals or real numbers.

There's more to it than that. Infinite sets are equivalent, and a good
question is as to why. It has to do with the fundament of the objects
in question, the domain of discourse, besides their very being as
infinite and thus unbounded in extent, i.e. there's always one more,
how that can be the case in light of Cantor's nested intervals and
diagonalization results. For the reals as a contiguous sequence, the
normal ordering of a segment of the reals is a well-ordering, and to
that sequence nested intervals doesn't apply. About diagonalization
versus Burali-Forti, that leads to dual and multiple representation,
from the existential character of the truly fundamental objects of the
theory, that is not necessarily obvious, and by the same token not
especially difficult to understand.

There are a variety of theories where there exist bijections between
given infinite sets and their powersets. Indeed, in some theories with
non-set ur-elements, which lead to quantification problems, the
powerset is smaller. In one sense the powerset is order type is
successor in the theory with the ubiquitous ordinals, or naturals.

As mathematics, or mathematical logic or structure within the
philosophy of mathematical logic is used to rigorously underlie any
science, it is of interest to many that their science have the
possibility of a rigorously consistent, complete, and perhaps even
concrete theory strong enough to support those scientific results.

To that end, there's one theory, the null axiom theory.

Ross

.

Relevant Pages

  • Re: Problem demonstrating that the set of binary strings is uncountable.
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  • Re: What does "Infinity" really mean?
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  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
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  • Re: On Well-Ordering(s) and Sets Dense in the Reals, Infinity
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