(Notations, non-standard concepts, and definitions used commonly in these investigations are detailed in this post.)
In a recent paper A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory, authors Adam Yedidia and Scott Aaronson argue upfront in their Introduction that:
“Like any axiomatic system capable of encoding arithmetic, ZFC is constrained by Gödel’s two incompleteness theorems. The first incompleteness theorem states that if ZFC is consistent (it never proves both a statement and its opposite), then ZFC cannot also be complete (able to prove every true statement). The second incompleteness theorem states that if ZFC is consistent, then ZFC cannot prove its own consistency. Because we have built modern mathematics on top of ZFC, we can reasonably be said to have assumed ZFC’s consistency.“
The question arises:
How reasonable is it to build modern mathematics on top of a Set Theory such as ZF?
1. “Like any axiomatic system capable of encoding arithmetic, …”
The implicit assumption here that every ZF formula which is provable about the finite ZF ordinals must necessarily interpret as a true proposition about the natural numbers is fragile since, without such an assumption, we can only conclude from Goodstein’s argument (see Theorem 1.1 here) that a Goodstein sequence defined over the finite ZF ordinals must terminate even if the corresponding Goodstein sequence over the natural numbers does not terminate!
2. “ZFC is constrained by Gödel’s two incompleteness theorems. The first incompleteness theorem states that if ZFC is consistent (it never proves both a statement and its opposite), then ZFC cannot also be complete (able to prove every true statement). The second incompleteness theorem states that if ZFC is consistent, then ZFC cannot prove its own consistency.”
The implicit assumption here is that ZF is -consistent, which implies that ZF is consistent and must therefore have an interpretation over some mathematically definable structure in which ZF theorems interpret as ‘true’.
The question arises: Must such ‘truth’ be capable of being evidenced objectively, or is it only of a subjective, revelationary, nature (which may require truth-certification by evolutionarily selected prophets—see Nathanson’s remarks as cited in this post)?
The significance of seeking objective accountbility is that in a paper, “The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas’ Gödelian Thesis“, which is due to appear in the December 2016 issue of Cognitive Systems Research, we show (see also this post) that the first-order Peano Arithmetic PA:
(i) is finitarily consistent; but
(ii) is not -consistent; and
(iii) has no ‘undecidable’ arithmetical proposition (whence both of Gödel’s Incompleteness Theorems hold vacuously so far as the arithmetic of the natural numbers is concerned).
3. “Because we have built modern mathematics on top of ZFC, we can reasonably be said to have assumed ZFC’s consistency.”
Now, one justification for such an assumption (without which it may be difficult to justify building modern mathematics on top of ZF) could be the belief that acquisition of set-theoretical knowledge by students of mathematics has some essential educational dimension.
If so, one should take into account not only the motivations of such a student for the learning of mathematics, but also those of a mathematician for teaching it.
This, in turn, means that both the content of the mathematics which is to be learnt (or taught), as well as the putative utility of such learning (or teaching) for a student (or teacher), merit consideration.
Considering content, I would iconoclastically submit that the least one may then need to accomodate is the following distinction between the two fundamental mathematical languages:
1. The first-order Peano Arithmetic PA, which is the language of science; and
2. The first-order Set Theory ZF, which is the language of science fiction.
A distinction that is reflected in Stephen G. Simpson’s more conservative perspective in Partial Realizations of Hilbert’s Program (6.4, p.15):
Finitistic reasoning (read ‘First-order Peano Arithmetic PA’) is unique because of its clear real-world meaning and its indispensability for all scientific thought. Nonfinitistic reasoning (read ‘First-order Set Thyeory ZF’) can be accused of referring not to anything in reality but only to arbitrary mental constructions. Hence nonfinitistic mathematics can be accused of being not science but merely a mental game played for the amusement of mathematicians.
(i) PA has two, hitherto unsuspected, evidence-based interpretations (see this post), the first of which can be treated as circumscribing the ambit of human reasoning about `true’ arithmetical propositions; and the second can be treated as circumscribing the ambit of mechanistic reasoning about `true’ arithmetical propositions.
It is this language of arithmetic—formally expressed as PA—that provides the foundation for all practical applications of mathematics where the latter could be argued as having an essential educational dimension.
(ii) Since ZF axiomatically postulates the existence of an infinite set that cannot be evidenced (and which cannot be introduced as a constant into PA, or as an element into the domain of any interpretation of PA, without inviting inconsistency—see paragraph 4.2 of this post), it can have no evidence-based interpretation that could be treated as circumscribing the ambit of either human reasoning about `true’ set-theoretical propositions, or that of mechanistic reasoning about `true’ set-theoretical propositions.
The language of set theory—formally expressed as ZF—thus provides the foundation for abstract structures that are only mentally conceivable by mathematicians (subjectively?), and have no physical counterparts, or immediately practical applications of mathematics, which could meaningfully be argued as having an essential educational dimension.
The significance of this distinction can be expressed more vividly in Russell’s phraseology as:
(iii) In the first-order Peano Arithmetic PA we always know what we are talking about, even though we may not always know whether it is true or not;
(iv) In the first-order Set Theory we never know what we are talking about, so the question of whether or not it is true is only of fictional interest.
The distinction is lost when—as seems to be the case currently—we treat the acquisition of mathematical knowledge as necessarily including the body of essentially set-theoretic theorems—to the detriment, I would argue, of the larger body of aspiring students of mathematics whose flagging interest in acquiring such a wider knowledge in universities around the world reflects the fact that, for most students, their interests seem to lie primarily in how a study of mathematics can enable them to:
(a) adequately abstract and precisely express through human reasoning their experiences of the world in which they live and work; and
(b) unambiguously communicate such abstractions and their expression to others through objectively evidenced reasoning in order to function to the maximum of their latent potential in acieving their personal real-world goals.
In other words, it is not obvious how how any study of mathematics that has the limited goals (a) and (b) can have any essentially educational dimension that justifies the assumption that ZF is consistent.