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Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on known independence results augmented by some idiomatically topological (as opposed to set-theoretic) argumentation. Thus one might prove a proposition on the assumption of the continuum hypothesis and then turn around and disprove it assuming (some strong but consistent form of) the negation of the continuum hypothesis.

But I have never seen work where Consist(ZFC), the prototypical independent axiom, takes the role that CH (or whatever) plays as described above. Thus I've never seen anyone establish a result in topology (or model-theoretic algebra, etc.) by first proving in (perhaps a strong but consistent extension of) ZFC+Consist(ZFC) and then disproving it in (perhaps a strong but consistent extension of) ZFC+$\neg$Consist(ZFC).

Question 1: Does the literature perhaps simply contain such an example that I've missed?

Question 2: If not, can one give metamathematical reasons why Consist(ZF) seems unsuited to such a role?

[I do realize that $\neg$Consist(ZFC) has a merely arithmetic character that might seem to leave it orthogonal to the concerns of a general topologist, but it does for example entail the existence of a model of ZFC with no internal model of ZFC, so it has its infinitistic face too.]

Question 3: I have heard it argued that an independence result derived from ZFC+$\neg$Consist(ZFC) would actually seem intrinsically bogus or unsatisfying on account of how we all actually believe the axiom $\neg$Consist(ZFC) false (even if we know can't prove it so from ZFC). Mathematicians seeking meaningful truths, the argument goes, shouldn't make it their business to study the implications of (even unprovably) false propositions lest they descend into sophistry. Can anyone offer a philosophical critique? To make myself clear, does there exist some qualitative sense in which $\neg$Consist(ZFC) $\implies P$ offers less insight into $P$ than, say $\neg CH\implies P$?

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2 Answers

Hi David,

I do not know of any results of the form you suggest.

There is a mathematical reason why these results are not too common, namely, the best known techniques to produce consistency results use either forcing axioms or inner model considerations. By a well-known absoluteness result of Shoenfield (see for example Section 13 of Kanamori's "The higher infinite"), all the models obtained this way satisfy the same $\Sigma^1_2$ statements. In particular, they satisfy the same arithmetic statements. This means that we cannot expect to arrive at models of $\lnot$Consist(ZFC) using these methods. But these are almost all the methods we have at our disposal!

Obtaining models with $\lnot$Consist(ZFC) or something similar, in addition to whatever interesting topological fact one is after, requires techniques for building (interesting) models that are not $\omega$-models, and would probably require that we leverage this strong ill-foundedness to our advantage. Until very recently there was no systematic approach to doing this, so the possibility, though interesting, was in essence intractable.

Recent work of Harvey Friedman suggests that this may change. At the moment, Friedman has used his techniques very effectively mostly to analyze certain combinatorial problems. His methods in a natural way require the construction of not-$\omega$-models (beginning with something like the negations of the combinatorial principle under study), and it is not too clear that one could do the same systematically with topological problems.

Friedman writes:

[...] it became clear that according to conventional wisdom, the Incompleteness Phenomena was confined to questions of an inherently set theoretic nature that was highly non concrete, and out of touch with normal mathematical activity.

[...] It was already clear to me at that time that despite the great depth and beauty of the ongoing breakthroughs in set theory regarding the continuum hypothesis and many other tantalizing set theoretic problems, the long range impact and significance of ongoing investigations in the foundations of mathematics is going to depend greatly on the extent to which the Incompleteness Phenomena touches normal concrete mathematics. This perception was confirmed in my first few years out of school at Stanford University with further discussions with mathematics faculty, including Paul J. Cohen.

[...] there were no candidates for Concrete Mathematical Incompleteness from ZFC being offered. In fact, to this day, no candidates for Concrete Mathematical Incompleteness have arisen from the natural course of mathematics.

[...] The second rationale for pursuing Concrete Mathematical Incompleteness preserves ZFC as the ambitious target. The idea is that normal mathematical activity up to now represents only an infinitesimal portion of eventual mathematical activity. Even if current mathematical activity does not give rise to Concrete Mathematical Incompleteness from ZFC, this is a very poor indication of whether this will continue to be the case, particularly far out into the future.

The tone, though cautions, is optimistic. The techniques Friedman has developed are presented in his book "Boolean relation theory", a draft of which can be found in Friedman's homepage. The quote above is from this book. Friedman's results typically require the construction of not-$\omega$-models, so studying his methods seems the most promising route towards realizing something like what you ask about.

Now, even in this case, his results do not typically land you at the level of $\lnot$Consist(ZFC), and it may require a couple of words to see their relevance.

A typical result has the form that certain combinatorial statement $P$ is equiconsistent with certain extension of ZFC by large cardinals. One direction is dome directly (even if the arguments are sophisticated): The combinatorics of the large cardinals allow us to obtain $P$.

However, $P$ tends to be of low complexity (say, it is arithmetic) so building from $\lnot P$ certain models where the consistency of some large cardinals fails requires that the models built are not $\omega$-models; it is here that several of Friedman's new techniques appear. In other words, we know in the standard model that $P$ holds, and it is in pathological models where the opposite is the case. To conclude, this indicates to me that even if not yet directly from current methods, these techniques seem the right direction to study if one hopes to eventually obtain a result of the kind as you suggest.

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If I remember correctly, Paris and Harrington proved that their statement does not follow from PA as it would otherwise imply the consistency of it. –  Péter Komjáth Jan 10 '11 at 8:34
Hi, Péter. Yes, unlike for ZFC, for theories weaker than PA there are a few techniques that would be helpful to accomplish the corresponding version of the question. PA is perhaps the best understood nowadays (and the Paris-Harrington result the first example of this phenomenon). –  Andres Caicedo Jan 10 '11 at 15:08
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I really like this question, and I think it gets to the heart of the study of large cardinals in set theory. To add to Andres's excellent answer, let me talk a little more about inner models and forcing extensions in the context of large cardinals. A number of set theorists like to assume the existence of large cardinals with the greatest consistency strength that is still believably relatively consistent with ZFC with the purpose of studying the "richest" set-theoretic universe possible. Such large cardinal notions are relatively consistent with statements such CH or $\lnot$CH, but are never relatively consistent with $\lnot CON(ZFC)$. Furthermore, while a large cardinal can be destroyed by moving to a forcing extension, ZFC models and their truth predicates will persist in all outer models including forcing extensions. On the flip side, moving to even the smallest inner model of G$\ddot{\textrm{o}}$del's Constructible Universe $L$ may very well dispose of truth predicates for some set models $M \in L$ of ZFC present in the original universe, but as Andres's post points out, there still must be some such $M$ with constructible truth predicates.

Now even under the assumption of large cardinals, you can of course move to set models of ZFC + $\lnot$CON(ZFC). But the point is that in the presence of sufficiently strong large cardinal hypotheses, these are not the reliable models from a philosophical point of view as they lack any of the richness of the ambient model. Since Andres mentioned Harvey Friedman, I'll mention that his brother Sy-David Friedman, has a stronger notion of consistency in the case that sufficiently powerful large cardinals exist called internal consistency (See e.g., Internal Consistency and the Inner Model Hypothesis ). In the presence of inner models with sufficiently strong large cardinals, a formula is internally consistent provided that it holds in an inner model. In this situation, it is never possible for $\lnot CON(ZFC)$ to be internally consistent, but it is possible that $\lnot CH$ is.

Let me now also provide another related link between large cardinals and your question. While we may not presently consider models of ZFC + $\lnot$CON(ZFC), we certainly do consider models of ZFC + $\lnot$LC for some large cardinal notion LC, and these give rise to very nice results (e.g., Covering lemmas).

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