> On 25 Feb 2019, at 20:35, Philip Thrift <cloudver...@gmail.com> wrote: > > via > https://twitter.com/JDHamkins/status/1100090709527408640 > > Joel David Hamkins @JDHamkins > > Must there be numbers we cannot describe or define? Definability in > mathematics and the Math Tea argument > Pure Mathematics Research Seminar at the University of East Anglia in Norwich > on Monday, 25 February, 2019. > > > Abstract: > > An old argument, heard perhaps at a good math tea, proceeds: “there must be > some real numbers that we can neither describe nor define, since there are > uncountably many real numbers, but only countably many definitions.
But that argument is rather weak, as the notion of cardinality is a relative notion, depending of the model (not the theory) that we might use. There are countable models of Cantor’s theory of set and the transfinite (cf the paradox of Skolem). If you agree o identify a real number with a total computable function (from N to N), as Turing did originally, then you can prove the existence of specific non definable real number, in any rich enough extension of any essentially undecidable theory. It is very simple, any theory reich enough to define a universal machine/number, is automatically essentially undecidable. It is a generator of infinitely many surprises for *any* machines and super-marching, etc. We know now that we know basically nothing, with such “rich” theories. Elementary arithmetic is already essentially undecidable. You can change the logic, and will get quite different view on the real numbers. In brouwer’s intuitionistic logic, and in the effective topos (which generalises Kleene’s realisability notion, and is based on the category of partial combinatory algebra): we have that all real number are computable, and all functions are continuous. I am not sure if we get that all real numbers will be definable, though. They might be not-not-definable ... > ” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of > pointwise definable structures in mathematics, structures in which every > object has a property that only it exhibits. A mathematical structure is > Leibnizian, in contrast, if any pair of distinct objects in it exhibit > different properties. x ≠ y ->. Ax ≠ Ay, that is Ax = Ay ->. x = y. That is the axiom of extensionality (in Combinators, lama calculus, set theories). I have used it in the elimination of variables in the combinations. It is a god’s gift, as it leads to simple efficacious combinators. It is, with the combinators, equivalent to [x](Ax) = A, when A has no free occurence of x. Not to be confuse with ([x]A)x = A (which is always true, and just defines what elimination of x means). > Is there a Leibnizian structure with no definable elements? Yes. The classical reals, or the classical set of total computable functions. > Must indiscernible elements in a mathematical structure be automorphic images > of one another? No. If we cannot discern them, we cannot build a morphism between them. I would say. > We shall discuss many elementary yet interesting examples, eventually working > up to the proof that every countable model of set theory has a pointwise > definable extension, in which every mathematical object is definable. > > http://jdh.hamkins.org/must-there-be-number-we-cannot-define-norwich-february-2019/ > > Lecture notes: > http://jdh.hamkins.org/wp-content/uploads/2019/02/Must-every-number-be-definable_-Norwich-Feb-2019.pdf I will take a look, but that does not make much sense, unless the logic is weakemed in some way. In some intuitionist set theory with choice, it might make sense. Bruno > > > - pt > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
