[FOM] Platonism and certainty

[Daniel Méhkeri]

I made a claim about Mahlo cardinals with respect to "set-theoretic 
platonists". Timothy Chow thought it might extend up to the 
measurables. I would like to state it a bit more precisely.

Is there anyone who has a high degree of certainty that CH, GCH, and 
indeed each sentence of first-order set theory is objectively true or 
objectively false, even if we don't know which, but who does not have a 
similar or greater degree of certainty that Mahlo cardinals exist? Is it 
correct that the level of the cumulative hierarchy where you cease to 
find a similar degree of certainty is the measurables?

One can ask an analogous question to "first-order number-theoretic 
platonists" about PA. But I am quite sure there is nobody who has a high 
degree of certainty that each sentence of first-order arithmetic is 
objectively true or objectively false, and does not have a similar or 
greater degree of certainty that PA is true.

In fact I can even ask the question even to "finitary platonists". Is 
there anyone who has a high degree of certainty that each Delta_0 
sentence involving primitive recursion functions is objectively true 
or objectively false, but does not have a similar or greater degree 
of certainty that epsilon_0 is well-founded?

"Well-founded" is meant in a restricted sense of course. Can we define
terms by descent recursion over certain computable orderings, similar
to the way we can define terms by primitive recursion, and what is the
order type that we can reach while still preserving certainty that they
are meaningful?

Regards,
Daniel Mehkeri


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