[FOM] reply to Harvey
martdowd at aol.com
martdowd at aol.com
Fri Jul 22 22:58:44 EDT 2016
Conservative new axiom theory provides some perspective on "absolute truth in set
theory", which has recently been discussed on the FOM newsgroup by Martin Davis
and Harvey Friedman; and in effect in a posting of Neil Barton of Apr 22.
There is some absolute truth in set theory, namely that there is a cumulative
hierarchy of sets, starting from the empty set, and proceeding by taking the
power set at successor stages and taking the union at limit stages. One introduces
some axioms and defines the ordinal numbers as prescribed by von Neuman, to "bootstrap"
a method for formalizing the iteration.
The absolute universe $V$ is the union of the sets $V_\alpha$ of the cumulative
hierarchy. According to ZFC, it satisfies the axiom scheme of replacement. In fact,
it satisfies second-order replacement. Since $V$ is a universe of discourse, the
endlessness of the cumulative hierarchy entails that these axioms reflect in some $V_\kappa$,
and $\kappa$ is an inaccessible cardinal.
Thus, new axioms can be added, in a conservative manner, and provide examples
of "absolute truth at work". Axioms of this type concern what Barton (op cit)
calls the "height" of the universe.
Martin Davis, in "Pragmatic Platonism", states
"if the iterative hierarchy is taken seriously, it (CH) does have a truth value
whether we can ever find it or not."
Absolute truth is not as forthcoming for CH and 0# as it is for the existence
of small large cardinals, such as the axiom of inaccessibles, Tarski's axiom that the
inaccessibles are unbounded, etc. These can be added to ZFC with a high degree of
confidence. Neither CH or ~CH, nor 0# or ~0#, can be. Perhaps this is one reason
set theorists have raised suspicions concerning absolute truth.
From: Martin Davis <martin at eipye.com>
To: fom <fom at cs.nyu.edu>
Sent: Thu, Jul 21, 2016 12:47 pm
Subject: [FOM] reply to Harvey
I’m pleased that Harvey has replied to my recent FOM post. I have a few comments.
Harvey has always refused to commit himself to a particular position on the philosophical issues that arise in connection with foundations. He has described himself in rather colorful language to emphasize his readiness to go wherever he finds an opening for serious work.
To begin with this message seems to reiterate this stance:
“The ‘matter of fact absolute truth’ (MOFAT) attitude to set theory is just one of many equally defensible positions, no more compelling than others.” But then he takes what seems to me a very decided position: he writes of “an inevitable shift” to a “better” or “best” way “to do set theory”. It seems to me that this “inevitable shift” is presented as the unique alternative to MOFAT. Personally, I find neither compelling. (My own view is based on a historical study of mathematics and the infinite:
The words “better, best” trouble me. It implies an opposing series “bad,worse, worst” and seems a departure from Harvey’s previous “anything goes” stance. Then, we are told that taking this position “sets the stage” for his current “hobby”, a novel approach to CH (one that I find fascinating). Of course someone who persists in a belief that CH may have a definite truth value can still find this work quite interesting, so there is really no need to "set the stage" in this way. Is Harvey wanting it made clear that he has no such belief? Is that why this project is a mere “hobby”?
Finally Harvey comments on my statement in my previous post: “because Pi-0-1 propositions can be expressed as a polynomial equation having no natural number solutions, a counter-example to a false Pi-0-1 proposition can be verified by a finite number of additions and multiplications of integers.” He comments, “Of course, this isn't sensitive to the nuances concerning what is meant by ‘can be verified’ here. It means computations that can be done using the algorithms we learned as children for adding and multiplying integers that will lead to the result 0. The only “nuance” I can imagine Harvey has in mind is that the numbers involved can be huge. But to take this “nuance” seriously would be to give up on number theory. If someone were to prove that the zeta function has a zero off the critical line, but the least such zero had an imaginary part greater than the number of atoms in the observable universe, wouldn’t we say that the Riemann Hypothesis had been disproved? Who would say to the contrary that a pragmatic Riemann Hypothesis had been proved because all zeros that we could hope to compute had been shown to be on the line? Similar conundrums could be presented to logicians who contemplate expressions in a formal system of arbitrary length.
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