[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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