Re: Logarithm of transfinite numbers

Tony Orlow wrote:
Listen, Dude. You ain't got no element you added to your set where it ever got
infinite, Pardner! I'm a startin' ta think yer set o' naturals ain't even big
as Texas.


Matt Gutting said:
No, it doesn't "get" infinite. It starts out infinite, by the axiom of induction.


Tony Orlow wrote:
What is the underlying justification for the axiom of induction? Please!

Axioms don't need any justification. They only need consistency with
the other axioms in the system.

And speaking as a Texan, yes, Texas is quite big, but it takes less
than 80 bits to number every molecule in the state. Compare that
to the 2^1100 or so bits needed to denumerate all the naturals up to
a googleplex. And compare _that_ to all the naturals countable with
a googleplex bits. Yet we're still a long way from exhausting the
naturals, of course, which requires a set size larger than any natural.

.

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