I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be consistent with just ZF, without choice? Let me be conservative, and say that by a "set of hyperreals," I just mean a set together with some relations and functions such that the transfer principle holds, and there exists $\epsilon > 0$ smaller than any real positive real number.
[Math] Is non-existence of the hyperreals consistent with ZF
model-theorynonstandard-analysisset-theory
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This came up yesterday in my Real Analysis course. I´m not sure if it is the sort of thing you are looking for and I´m also not sure if the use of $AC$ is essential, but….
Suppose $X$ is a complete metric space and $\{E_n:n\in \omega\}$ is a nested sequence of non-empty closed subsets of $X$ such that $\lim_{n \to \infty} \operatorname{diam} (E_n)=0$. Then there is a point $p$ that belongs to every $E_n$.
The only proof I know of this uses countable choice to get a sequence $\{ x_n\}$ with $x_n \in E_n$; then this sequence is a Cauchy sequence, etc.
But then the object whose existence you are trying to prove (i.e. $p$) is unique, by the condition on the diameters.
As part of his question, Bell Crowell correctly observes:
"Section 9 of the Ehrlich paper discusses the relationship between R∗ and No within NBG. He presents Keisler's axioms for the hyperreals, which basically say that they're a proper extension of the reals, the transfer principle holds, and they're saturated. At the end of the section, he states a theorem: "In NBG [with global choice] there is (up to isomorphism) a unique structure ⟨R,R∗,∗⟩ such that [Keisler's axioms] are satisfied and for which R∗ is a proper class; moreover, in such a structure R∗ is isomorphic to No.""
At that time I made it absolutely clear that the first part of the result is due to H.J. Keisler (1976) and that my modest contribution is to point out the relation (as ordered fields) between R* and No. The work of Keisler and the relation of my work to it seem to be lost in the remarks of Vladimir.
Of course, attributing the result to Keisler, as I remain entirely confident I correctly did, does not diminish the subsequent important contributions of others.
Edit: Readers interested in reading the paper including the discussion of Keisler's work may go to: http://www.ohio.edu/people/ehrlich/
EDIT: Since Vladimir appears to insist in his comment below that Keisler DOES NOT discuss proper classes in 1976, I am taking the liberty to quote Keisler and some of the relevant discussion from my paper. I will leave it to others to decide if I am giving Keisler undue credit.
Following his statement of his Axioms A-D of 1976--the function axiom, the solution axiom, and the axioms the state that R* is proper ordered field extension of the complete ordered field R of real numbers--Keisler writes:
“The real numbers are the unique complete ordered field. By analogy, we would like to uniquely characterize the hyperreal structure ⟨R,R∗,∗⟩ by some sort of completeness property. However, we run into a set-theoretic difficulty; there are structures R* of arbitrary large cardinal number which satisfy Axioms A-D, so there cannot be a largest one. Two ways around this difficulty are to make R* a proper class rather than a set, or to put a restriction on the cardinal number of R*. We use the second method because it is simpler.” [Keisler 1976, p. 59]
With the above in mind, Keisler sets the stage to overcome the uniqueness problem by introducing the following axiom, and then proceeds to prove the subsequent theorem.
AXIOM E. (Saturation Axiom). Let S be a set of equations and inequalities involving real functions, hyperreal constants, and variables, such that S has a smaller cardinality than R*. If every finite subset of S has a hyperreal solution, then S has a hyperreal solution.
KEISLER 1 [1976]. There is up to isomorphism a unique structure ⟨R,R∗,∗⟩ such that Axioms A-E are satisfied and the cardinality of R* is the first uncountable inaccessible cardinal.
If ⟨R,R∗,∗⟩ satisfies Axioms A-D, then R* is of course real-closed. It is also evident that, if ⟨R,R∗,∗⟩ further satisfies Axiom E, then R* is an $\eta_{\alpha}$-ordering of power $\aleph_{\alpha}$, where $\aleph_{\alpha}$ is the power of R*. Accordingly, since (in NBG) No is (up to isomorphism) the unique real-closed field that is an $\eta_{On}$-ordering of power $\aleph_{On}$, R* would be isomorphic to No in any model of A-E that is a proper class (in NBG).
Motivated by the above, in September of 2002 we wrote to Keisler, reminded him of his idea of making “R* a proper class rather than a set”, observed that in such a model R* would be isomorphic to No, and inquired how he had intended to prove the result for proper classes since the proof he employs, which uses a superstructure, cannot be carried out for proper classes in NBG or in any of the most familiar alternative class theories.
In response, Keisler offered the following revealing remarks, which he has graciously granted me permission to reproduce.
"What I had in mind in getting around the uniqueness problem for the hyperreals in “Foundations of Infinitesimal Calculus” was to work in NBG with global choice (i.e. a class of ordered pairs that well orders the universe). This is a conservative extension of ZFC. I was not thinking of doing it within a superstructure, but just getting four objects R, R*, <*, * which satisfy Axioms A-E. R is a set, R* is a proper class, <* is a proper class of ordered pairs of elements of R*, and * is a proper class of ordered triples (f,x,y) of sets, where f is an n-ary real function for some n, x is an n-tuple of elements of R* and y is in R*. In this setup, f*(x)=y means that (f,x,y) is in the class *. There should be no problem with * being a legitimate entity in NBG with global choice. Since each ordered triple of sets is again a set, * is just a class of sets. I believe that this can be done in an explicit way so that R, R*, <*, and * are definable in NBG with an extra symbol for a well ordering of V." [Keisler to Ehrlich 10/20/02]
Moreover, in a subsequent letter, Keisler went on to add:
I did not do it that way because it would have required a longer discussion of the set theoretic background. [Keisler to Ehrlich 5/14/06]
Best Answer
The answer is yes, provided ZF itself is consistent. The reason is that the existence of the hyperreals, in a context with the transfer principle, implies that there is a nonprincipal ultrafilter on $\mathbb{N}$.
Specifically, if $N$ is any nonstandard (infinite) natural number, then let $U$ be the set of all $X\subset\mathbb{N}$ with $N\in X^*$. This is a nonprincipal ultrafilter on $\mathbb{N}$, since:
So $U$ is a nonprincipal ultrafilter on $\mathbb{N}$. The way that I think about $U$ is that it concentrates on sets that express all and only the properties held by the nonstandard number $N$. (See also my answer to A remark of Connes, where I make a similar point, and explain that, therefore, nonstandard analysis with the transfer property implies that there must be a non-measurable set of reals.)
Thus, in a model of ZF with no nonprincipal ultrafilter on $\mathbb{N}$ (and as Asaf mentions in the comments, there are indeed such models if there are any models of ZF at all), there is no structure of the hyperreals satisfying the transfer principle.