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None of this is somehow secret. The standard name for this is "definable"[0]. Although, one has to be really careful with this sort of thing; there are appare

None of this is somehow secret. The standard name for this is "definable"[0]. Although, one has to be really careful with this sort of thing; there are apparently a number of subtle logical issues[1] that come up when talking about these... (Note, by the way, that there's any number of other fields one could put inbetween; such as the field of algebraic reals, or computable reals, or the fraction field of the ring of periods...) [0] https://en.wikipedia.org/wiki/Definable_real_number https://en.wikipedia.org/wiki/Definable_real_number [1] https://mathoverflow.net/a/44129/5583 https://mathoverflow.net/a/44129/5583

dwohnitmok 7y ago
Well that Math Overflow post is excellent. One of the logical issues is that there is a model of ZFC where all reals are definable/useful. I'm guessing that's not what the author of this blog post is going for... If this seems impossible given that the number of definitions is countable, note first that it is possible that a model of ZFC is itself countable (in a larger ambient model), but it cannot witness the countability of sets within itself. So when we say that a set is uncountable in ZFC, it is sometimes useful to make the distinction that it is only uncountable in the implicit model under discussion. Then note that definability, unlike countability, cannot be itself defined in the language of ZFC (due to Tarski's undefinability of truth result). Note that this is different from saying it's independent of ZFC. It cannot even be expressed in ZFC. Hence, unlike countability, there is no "relative" concept of definability, at least not relative to first-order ZFC. Therefore the statement "every element of this model is definable" is more absolute than "every element of this model is countable" (but not absolutely absolute, we still have an ambient model we're working in, just a richer theory for that model). The usual diagonalization argument within our entirely definable model of ZFC to try to construct a definable real number not contained in any countable enumeration of definable real numbers fails because we have no enumeration of definable real numbers. This is not a failure of constructivism (it is ZFC after all, we do have choice), but rather a consequence of the fact that definability cannot be expressed in ZFC so we don't have a way of even talking about the set of all definable real numbers within our model.
- dwohnitmok 7y ago
  Upon re-reading my reply, it might be worthwhile to simply quote Hamkins' conclusion in full. > And therefore neither are you able to do this in general. The claims made in both in your question and the Wikipedia page [the Wikipedia page has now since been updated] on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable, and every object is uniquely specified by a property.

dwohnitmok 7y ago

Well that Math Overflow post is excellent. One of the logical issues is that there is a model of ZFC where all reals are definable/useful. I'm guessing that's not what the author of this blog post is going for... If this seems impossible given that the number of definitions is countable, note first that it is possible that a model of ZFC is itself countable (in a larger ambient model), but it cannot witness the countability of sets within itself. So when we say that a set is uncountable in ZFC, it is sometimes useful to make the distinction that it is only uncountable in the implicit model under discussion. Then note that definability, unlike countability, cannot be itself defined in the language of ZFC (due to Tarski's undefinability of truth result). Note that this is different from saying it's independent of ZFC. It cannot even be expressed in ZFC. Hence, unlike countability, there is no "relative" concept of definability, at least not relative to first-order ZFC. Therefore the statement "every element of this model is definable" is more absolute than "every element of this model is countable" (but not absolutely absolute, we still have an ambient model we're working in, just a richer theory for that model). The usual diagonalization argument within our entirely definable model of ZFC to try to construct a definable real number not contained in any countable enumeration of definable real numbers fails because we have no enumeration of definable real numbers. This is not a failure of constructivism (it is ZFC after all, we do have choice), but rather a consequence of the fact that definability cannot be expressed in ZFC so we don't have a way of even talking about the set of all definable real numbers within our model.

dwohnitmok 7y ago
Upon re-reading my reply, it might be worthwhile to simply quote Hamkins' conclusion in full. > And therefore neither are you able to do this in general. The claims made in both in your question and the Wikipedia page [the Wikipedia page has now since been updated] on the existence of non-definable numbers and objects, are simply unwarranted. For all you know, our set-theoretic universe is pointwise definable, and every object is uniquely specified by a property.