[FOM] strong hypotheses and the theory of N

Aatu Koskensilta Aatu.Koskensilta at uta.fi
Sun Mar 14 21:48:06 EDT 2010

(Note to moderator: this message corrects a slight oversight in the  
earlier message. Please disregard the previous version.)

Quoting Monroe Eskew <meskew at math.uci.edu>:

> It would seem a reasonable requirement that all strong hypotheses
> which set theorists explore or use should all agree on the theory of
> natural numbers.  So then how do we know whether whatever large
> cardinal, forcing axiom, determinacy statement, etc. we're looking at
> will not say anything about omega that a different such hypothesis
> contradicts?

Since the calibration of the strength of naturally occurring set  
theoretic principles is done by means of forcing and inner models,  
neither of which disturbs truth of arithmetical statements, if such a  
principle is consistent it's arithmetically compatible with all  
equally strong or weaker principles. A statement the strength of which  
can't be calibrated in the usual hierarchy using these techniques is  
regarded with suspicion.

Aatu Koskensilta (aatu.koskensilta at uta.fi)

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

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