On 04/05/2013 6:04 PM, fom wrote: > On 5/4/2013 11:07 AM, Frederick Williams wrote: >> Nam Nguyen wrote: >>> >>> On 26/04/2013 11:09 AM, Nam Nguyen wrote: >> >>>> On 2013-04-25, FredJeffries <email@example.com> wrote: >>>>> >>>>> Now PA has been proved consistent in ZF or NBG, but then that >>>>> brings the consistency of axioms for set theory. >>> >>> Exactly right. And exactly my point. >>> >>> Somewhere, somehow, a circularity or an infinite regression >>> of _mathematical knowledge_ will be reached, >> >> How does one reach an infinite regression? >> >>> and at that point >>> we still have to confront with the issue of mathematical relativity. >> >> It is not the case that either we go round in a circle or we regress >> forever. > > Out of curiosity, how do you come to that conclusion? I have > come to the exact opposite conclusion.
> The only sense I can > make of foundations is that it is more like a jigsaw puzzle > that must address circularity and regress directly and with > the objective of making it harmless.
I couldn't agree less; and that is exactly what I've proposed for the past years.
The way out of, or the way to manage and address, the circularity or and the regression mentioned above is to accept certain non-inference rules about unknowability, impossibility, as part of the FOL reasoning edifice.
The entire human mathematical reasoning would then be a balance of what we can know, through the canonical rules of inference, and what we can _not_ know, through the newly accepted _non-inference rules_ .
Exactly what these rules of non-inference are we can sort it out. But there's a consequence we have to accept as well: mathematics in general would be relativistic, with certain mathematical truth values can be chosen _by choice_ (at will).
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.