Re: Axiom of infinity and the set of all hereditary finite sets.

On Sep 29, 12:58 pm, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> writes:

The set of all hereditarily finite sets, comprised only of
hereditarily finite sets, would be a hereditarily finite set, and thus
contain itself, similarly to how the set of all ordinals would be an
ordinal and contain itself. (?)

The set of hereditarily finite sets is not a hereditarily finite set.

I agree.

The set of all hereditarily finite sets is an infinite set.
Since if it is finite, then it would be hereditarily finite
and it would contain itself as a member and this would
violate regularity, and thus by negation it is not finite.
Now the set which has the set of all hereditarily finite sets
as its member is finite but not hereditarily finite, thus
it is not a member of the set of all hereditarily finite sets
thus it doesn't suffer the same problem as that of the class of all
finite sets sufferes in ZFC ( this later class is not a set in ZFC).

For more information see my new alternative axiomatization of ZFC
that I have posted in anther topic in this usenet.

http://groups.google.com/group/sci.logic/browse_frm/thread/7ec7ea91363316bf

This system of axiomatization proves the existence of the set of all
hereditary finite sets, as an infinite set, and it replaces the axiom
of infinity, as well as the axioms of pairing, union and power
also it replaces the schemas of separation and replacement.

Zuhair

(Consider Goedelian incompleteness interpreted as that there are no
absolute truths about the natural integers. Is that not an absolute
truth about the natural integers?)

One might just as well consider Gödelian incompleteness interpreted as
the claim that bananas are tastier than broccoli.

--
Aatu Koskensilta (aatu.koskensi...@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus


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