[FOM] Infinity and the "Noble Lie"
Aatu Koskensilta
aatu.koskensilta at xortec.fi
Tue Dec 13 09:57:15 EST 2005
On Dec 10, 2005, at 10:59 PM, joeshipman at aol.com wrote:
> The point of the question is that I don't expect everyone to have the
> same answer as you. What is it about the large cardinals up to a weakly
> compact cardinal that makes you believe they are true?
The reasoning is somewhat involved, but it seems that a weakly compact
cardinal is as far as one can get by extending ZFC with a hierarchy of
predicative proper classes (definable by iterating truth predicate for
the language of set theory along (possibly class sized) well-orderings)
and adding predicatively meaningful reflection principles. I accept the
axioms of ZFC as true for the rather uninteresting reason that I find
it difficult to imagine how they could be false. Adding a hierarchy of
predicative proper classes is "epistemologically conservative" over ZFC
if one accepts legitimacy of the notion of set theoretic truths.
Reflection principles aren't conservative in this sense, but I find
them (or, rather, the informal idea from which various exact principles
more or less naturally follow) easily acceptable on the somewhat
dubious grounds of wanting as many interesting structures to appear in
the cumulative hierarchy as possible.
There is an underlying tension in my haphazard approach: on the other
hand I evaluate various set theoretic axioms from the point of view of
how well they cater for the need of a unified framework for
mathematics, i.e. as a matter of stipulation more than of fact, yet my
other hand is very concerned about questions of fact - this perhaps
reflects the dual role of set theory as a) a framework for mathematics
and b) a discipline of mathematics. Be that as it may, I'm always
willing to unreservedly assert of what ever axiom I accept that it is
true, regardless of whether I came to accept it as a result of
stipulation or by more serious pondering.
Aatu Koskensilta (aatu.koskensilta at xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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