User flash sheridan - MathOverflow

Answer by Flash Sheridan for Can we have A={A} ?

2013-05-09T16:18:39Z

(This is intended as a comment on Thomas Forster’s point above, “i think there is a literature about them that goes back earlier than Quine,” but I’m over the character limit.)

Much earlier than Quine; Frege uses the trick in §10 of the Grundgesetze, setting (loosely, in modern terms) the True and False to their own singletons. He considers using the trick in greater generality, à la Quine, in footnote 17: “A natural suggestion is to generalize our stipulation so that every object is regarded as a course-of-values, viz., as the extension of a concept under which it and it alone falls.” I would be stunned if this did not influence Quine’s use, either directly or indirectly, subconsciously or consciously.

Heck discusses the stipulation is great detail in Part I of Reading Frege’s Grundgesetze, as part of his treatment of The Julius Caesar Problem, i.e., how can we be sure that the extension of some concept is not equal to Julius Caesar?

Personally, I feel that it is still too soon for Frege to worry about The Julius Caesar Problem. As he pointed out elsewhere, discussing a language that hasn’t yet been fully formalized is fraught with peril, and “Julius Caesar” is not yet a precisely-defined term in a formal language. (Perhaps Caesar will end up being formalized as the extension of a concept after all—I don’t want to pre-judge the formalization of zoology.) And obviously his proof of referentiality in §10 has to break down somewhere, or it would have been a consistency proof for naïve set theory.

Answer by Flash Sheridan for Consistency of the concept of the collection of all collection

2013-03-25T15:41:00Z

Church’s “Set Theory with a Universal Set” (see, e.g., Thomas Forster’s article) has a set of all sets, and Church provides a model (actually an interpretation in ZFGC) for it. He never published the actual consistency proof, though, and after looking in the Church archives at Princeton, I suspect that he abandoned it, as well as a couple of more complicated theories in which he was attempting to converge with New Foundations. I provide a full consistency proof in an article I’ve submitted to Logique et Analyse, for a variant in which the singleton function is a set (which is impossible in New Foundations), though the natural generalization leads to a variant of the Russell Paradox, the set of all non-self-membered sets equinumerous to the universe. Arnold Oberschelp also has a set theory with a universal set and the singleton function, though his consistency proof is difficult to verify, as a key part is merely a reference to an earlier article which uses a significantly different formalism.

Answer by Flash Sheridan for mutually incompatible abstraction terms?

2011-09-11T17:26:50Z

For a philosophical perspective on this, see Alan Weir’s “Neo-Fregeanism: An Embarrassment of Riches”: “The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles[;] thus not all consistent abstractions can be true.”

Closer to my own work, Oberschelp [1973] and I [forthcoming] have consistency proofs for set theories with a universal set in which the singleton function is a set. This would lead to an immediate contradiction in Quine’s New Foundations, whose consistency is an open problem. Another open problem is constructing a model in ZFC for the combination of Church’s set theory with a universal set (which has a sequence of Frege-Russell cardinals for equivalence relations generalizing equinumerosity), with Mitchell’s variant, which lacks the Frege-Russell cardinals but has an unrestricted axiom of power set. My conjecture is that such a construction would be impractical with current techniques, but we’d all be very disappointed if the combined theory were inconsistent.

Bibliography

• Alonzo Church 1974a. “Set Theory with a Universal Set,” Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics XXV, ed. Leon Henkin, American Mathematical Society pp. 297-308. (Delivered 24 June 1971.)

• Emerson Mitchell 1976. A Model of Set Theory with a Universal Set, unpublished Ph.D. thesis, University of Wisconsin at Madison.

• Arnold Oberschelp 1964a. “Eigentliche Klasse als Urelemente in der Mengenlehre,” Mathematische Annalen 157 pp. 234-260. [MR 31#2136]. (Delivered 20 August 1962)

• Arnold Oberschelp 1964b. “Sets and Non-Sets in Set Theory” (abstract), received 3 June 1964, Journal of Symbolic Logic XXIX p. 227

• Arnold Oberschelp 1973. Set Theory over Classes, Dissertationes Mathematicæ (Rozprawy Mat.) 106. [MR 42 #8300].

• Ulf Friedrichsdorf 1979. “Zur Mengenlehre über Klassen,” Zeitschrift f. Mathem. Logik 25, pp. 379-383. Contains helpful summary of [Oberschelp 1964a & 1973].

Note that the crucial part of the consistency proof in both [Friedrichsdorf 1979] and [Oberschelp 1973] is merely a reference to [Oberschelp 1964a], which uses a very different formalism.

• Alan Weir 2003. “Neo-Fregeanism: An Embarrassment of Riches,” Notre Dame Journal of Formal Logic volume 44, Number 1 (2003), pp. 13-48, http://projecteuclid.org/euclid.ndjfl/1082637613

Answer by Flash Sheridan for Best tablet computer for mathematics

2011-06-30T17:02:42Z

To address part of your question, the iPad is far more than a toy for mathematicians, especially for reading mathematical articles. Goodreader is the best means of reading and annotating PDFs I’ve ever seen; in particular, the usual size of LaTeX articles fits very well on the screen, with a little zooming, even in portrait mode. I’ve virtually eliminated my use of my printer, both for reading others’ articles and correcting my own XeLaTeX output.

Be very sure to test a PDF reader on real articles before it’s too late; even Adobe renders the Computer Modern font unacceptably on-screen, though with decidedly different results in different situations and platforms, and Okular on Linux is virtually unreadable at 100% zoom.

For note-taking, I’m not a big fan of raw handwritten text (disclaimer: I worked at Apple during the Newton days, and at Palm); I expected to have to buy a hardware keyboard for my iPad, but haven’t needed to yet.

Answer by Flash Sheridan for Physics and Church–Turing Thesis

2011-03-22T19:22:47Z

To follow up on Michael Beeson’s answer (I’m not allowed to post comments yet), Robin Gandy’s paper was “Church's Thesis and Principles for Mechanisms” in The Kleene Symposium (http://dx.doi.org/10.1016/S0049-237X(08)71257-6). He established that “Turing's analysis of computation by a human being does not apply directly to mechanical devices,” provided a “set-theoretic form of description for discrete deterministic machines,” and “proved that if a device satisfies the principles then its successive states form a computable sequence.”

The most surprising point is that the result depends on non-Newtonian physics, in particular the impossibility of instantaneous action at a distance. The term “Gandy machine” is sometimes used for this variant of Turing machine, e.g. by Blass and Sieg; the notion even has its own entry in the Esoteric Programming Languages wiki.

(Disclaimer: Robin was my supervisor, and Turing was his, so I’m hardly unbiased.)

Answer by Flash Sheridan for Set theory and alternative foundations

2011-03-19T01:10:45Z

To follow up further on Joel David Hamkins's answer on geometry, Frege’s last work (two despairing decades after Russell’s Paradox demolished his Grundgesetze der Arithmetik) was a brief unpublished paper entitled “Neuer Versuch der Grundlegun der Arithmetik,” based on geometry with “the final goal, the general complex numbers.” (As in the Grundgesetze, he emphasizes that real numbers are ratios of quantities, not quantities themselves.)

Answer by Flash Sheridan for How to define tuples?

2011-03-18T16:53:49Z

Some people do have to care about such details, at least in unusual contexts, and I do think it’s generally worth being aware of your foundations. The details of the definition of ordered pairs is crucial in Quine’s New Foundations (e.g., http://en.wikipedia.org/wiki/New_Foundations#Ordered_pairs), and taking it as primitive can have actual set-theoretic consequences in NF. In Church’s unpublished supplement to his “Set Theory with a Universal Set,” he uses a deliberately ugly [my interpretation] definition of m-tuple to avoid collisions. In my follow-on work, I use the usual Kuratowski definition of ordered pairs, since their internal structure allowed me to model the singleton function as a set, since it’s a 2-equivalence class, for a generalization of Church’s definition of j-equivalence relations.

Answer by Flash Sheridan for How much of ZFC does Quine's New Foundations prove?

2011-03-12T00:13:24Z

While I second (with highly biased motivation) the recommendation of Forster’s book, for questions like this an easier starting point than Forster’s book or Holmes’ articles might be Holmes’ book Elementary Set Theory with a Universal Set, originally volume 10 of the Cahiers du Centre de logique, with a corrected online version at http://math.boisestate.edu/~holmes/holmes/head.ps.

(This should perhaps have been a comment to the original answer, but I don’t have enough reputation points yet to post comments.)

Answer by Flash Sheridan for Intuitive and/or philosophical explanation for set theory paradoxes

2011-03-11T06:23:20Z

To get back to the original question, I blame half-hearted Platonism for the paradoxes: If you’re enough of a mathematical Platonist to believe that the universal set is a completed totality, then it’s hypocritical to believe that you can later construct new sets via a comprehension scheme. That does of course leave open the question of what sets other than the universal set exist; the best answer I know of is Alonzo Church’s “Set Theory With a Universal Set,” which has an axiom of complements and a generalization of Frege-Russell cardinals as sets, and is equiconsistent with ZF.

Disclaimers: Church was cagey about the philosophical motivation behind his theory; Thomas Forster (in Oxford Logic Guides 20 & 31, also entitled Set Theory With a Universal Set,) likened my reasoning to “the grating sound of a virtue being made of necessity,” though I did come up with it before I discovered Church’s theory, in an essay which R.I.G. Hughes gave a D.

Church’s original paper is heavy going, and omits the consistency proof. There are some unpublished lecture notes of his proof for the case m=0, but they’re even heavier going, and the real challenge is for m>0. Forster’s book is probably the best place to start on Church’s theory (and related theories by Emerson Mitchell and me), though the first edition has more detail than the second; see also http://www.dpmms.cam.ac.uk/~tf/church2001.pdf.

Comment by Flash Sheridan

2013-04-01T14:43:51Z

Also note that considerable simplification can be achieved in some set-theoretic definitions by, rather than creating a special case, classifying the empty set as a limit ordinal, i.e., the limit of all ordinals which do not exist.

Comment by Flash Sheridan

2012-04-16T12:05:30Z

@Lee: Francois’s URL works, but the MathOverflow parser mangled the link. en.wikipedia.org/wiki/Curry%27s_paradox may have better luck.

Comment by Flash Sheridan

2011-09-01T23:15:54Z

Part II I believe someone eventually derived a full contradiction from Frege’s hasty Way Out, but I can’t find the reference. I think he did genuinely give up, especially on deriving arithmetic from logic. See, for example, his late abortive attempt to derive it from geometry, “A New Attempt at a Foundation for Arithmetic,” in the Posthumous Writings.

Comment by Flash Sheridan

2011-09-01T23:14:55Z

A few corrections about Frege’s Grundgesetze, Part I of II. He didn’t finish the work, he merely sent the appropriate number of pages to the printer. The last section of volume II (§245, p. 243) discusses “Die nächste Aufgabe,” “the next problem.” See Dummett’s Frege: Philosophy of Mathematics, p. 241; he estimates that Volume II contains only about three quarters of Part III.

Comment by Flash Sheridan

2011-03-11T22:00:36Z

I seem to have run into what I’ll call Flash’s Paradox, of the set of all non-self-membered sets equinumerous to the universe; I can hope that a similar extension of Church’s original theory would avoid the difficulty.

Comment by Flash Sheridan

2011-03-11T21:59:36Z

Admittedly the standard formalism does possess impressively unreasonable effectiveness, and the various programs of “Fixing Frege” (e.g., also as a basis for arithmetic) are still well short of the dominant paradigm. My own attempt to extend Church’s theory (with the singleton function as a set), though it’s equiconsistent with ZF as far as it goes, seems unlikely to be able to go far enough for the needs of category theory (e.g., Feferman’s “Enriched stratified systems for the foundations of category theory,” in What is Category Theory? [G. Sica 2006]).

Comment by Flash Sheridan

2011-03-11T21:58:28Z

The discussion here has also been what I call “temporalist,” e.g., “Eventually,” “new a's from old” [quoting Scott], and “already.” One comment on the temporalist metaphor and predicativity, which is not original with me (I heard it from someone at Oxford in the 80’s): If you’re serious about the metaphor, why can sets at one stage be defined via quantification over later stages?

Comment by Flash Sheridan

2011-03-11T21:56:09Z

“Iterative” isn’t my metaphor, so I’m not sure I’m the right person to defend it, but even Gödel, in the classic Platonist defensive of the iterative conception, uses explicitly temporal language at least once: “immediately,” [footnote 12 of “What is Cantor’s Continuum Problem,” reprinted in Benacerraf & Putnam]. Boolos uses temporal language emphatically in “The Iterative Conception of Set,” e.g., “now” twice, once italicized, on page 491 of B&P. He does of course disclaim the metaphor for the formal exposition, but it’s the motivation we’re discussing now, not the formalism.