1
$\begingroup$

In discussions about the hyperreals where the context seems to be that a first-order theory of the reals has been extended to a first-order theory of the hyperreals (obeying the transfer principle), the definition of the floor function always seems to be taken as a given when the hyperintegers are discussed, whereas the hyperintegers are treated as something that needs to be defined in terms of the floor function instead of the other way around.

For example, on the Wikipedia page for the hyperintegers,

The standard integer part function: ⌊x⌋ is defined for all real x and equals the greatest integer not exceeding x. By the transfer principle of nonstandard analysis, there exists a natural extension: ∗⌊⋅⌋ defined for all hyperreal x, and we say that x is a hyperinteger if x = ∗⌊x⌋. Thus, the hyperintegers are the image of the integer part function on the hyperreals.

However, the floor function cannot be defined in a first-order theory of the reals which doesn't have the integers in its vocabulary, otherwise the integers would be definable in a first-order theory of the reals which infamously they are not.

Therefore, to get to the hyperreals and then the hyperintegers from a first-order theory of the reals you could either add the construction of ℤ or ⌊⋅⌋ to however you constructed the reals for your theory, so that your theory has ℤ or ⌊⋅⌋ in its vocabulary. If you chose ℤ then ℤ goes on to represent the hyperintegers once you've turned your reals into hyperreals. If you chose ⌊⋅⌋ then you define the (hyper)integers as Wikipedia does above.

It seems to me that these are equivalent but every discussion I see chooses ⌊⋅⌋ and doesn't even say that it has to be added to the vocabulary of the first-order theory, they just treat the existence of ⌊⋅⌋ as a given and then go on to use it to define ℤ. Why isn't ℤ treated as a given and used to define ⌊⋅⌋? They're both undefinable in first-order theories of the reals and thus need to be constructed along with the reals to be in the vocabulary of the theory, right?

Thanks in advance!

$\endgroup$
3
  • 1
    $\begingroup$ Wikipedia's source is Keisler, and Keisler defines the floor of $x$ as "the greatest integer $n$ such that $n\leq x$" (emphasis mine). So the existence of integers is being presupposed here. As to why Keisler uses the floor function instead of just forming ${}^*\mathbb Z$, it seems that he is avoiding talking about $*$-transforms of sets directly; his statement of the Transfer principle is "Every real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions". $\endgroup$ Commented Oct 20 at 23:49
  • $\begingroup$ He then goes on and seems to say that the "hyperreal natural extension" of a real function is what you get by replacing "real" with "hyperreal" in any defining statement of that function. $\endgroup$ Commented Oct 20 at 23:51
  • $\begingroup$ @NicholasTodoroff thank you! $\endgroup$ Commented Oct 21 at 9:19

1 Answer 1

Reset to default
0
$\begingroup$

It's just a question of taste. Every real-valued function on the reals and every subset of the reals has a hyperreal extension. This is true whether or not the function is continuous or definable. So, in particular, we could simply represent the sets with their characteristic functions, which all have hyperreal extensions.

In the construction of the hyperreals, to make the transfer principle work, in fact every function needs to have a symbol added to the language of the original theory, before the ultrapower is formed. Few low-level texts emphasize this. They talk about how an individual function can be extended into the ultrapower, but not about how a symbol for the function has to be included in the language in order to apply the transfer principle. I would say this is because most elementary presentations try to avoid talking too much about first-order logic and syntax.

Section 4 of these notes has a relatively careful analysis of the transfer principle and the structure that is used to make the ultraproduct: https://math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Davis.pdf . Note that unary relations are the same as sets.

$\endgroup$
4
  • $\begingroup$ I can imagine some pedagogical reasons why, in elementary calculus, it is easier to talk about extending a function to a larger domain, compared to extending a set like $\mathbb{Z}$ to a new set ${}^*\mathbb{Z}$. Students are used to the idea of extending a partial function to a larger domain, but think of sets as particular collections of elements. $\endgroup$ Commented Oct 21 at 10:03
  • $\begingroup$ Thank you, I've only just skimmed those notes but they look like exactly the sort of thing I'm looking to read at the moment, and each part of your answer makes a lot of sense and was what I was looking for, I feel confident on the matter now, so thanks again! (I don't seem to have upvoting powers though.) Out of interest, how do people such as yourself have notes like those to hand? Was it something you'd read before and recalled, or is finding this sort of thing just easy when you're practiced at that sort of thing and knowledgeable in the field? $\endgroup$ Commented Oct 21 at 14:41
  • $\begingroup$ @Undefinable_Axiom - in this case I did find the notes online. I knew I was looking for a construction of the hyperreals that explicitly discussed the expansion of $\mathbb{R}$ with all real functions, but didn't know where it would be written up. I found a number of documents and this seemed the best of them for the issue at hand $\endgroup$ Commented Oct 21 at 15:22
  • $\begingroup$ Well thank you, I really appreciate it. For what it's worth I looked through your account here and I like your style. Sorry for the possibly stupid question and for possibly taking up more of your time but how did you go about searching for notes that did that? Google? What roughly were your search terms if so? $\endgroup$ Commented Oct 21 at 16:33

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.