These exercises (due September 28) are mostly meant to test your understanding of compactness.

Let be a nonstandard model of Show:

(Overspill) Suppose that is definable (with parameters) and that Show that is finite.

(Underspill) Suppose that is definable and that Show that there is some infinite such that all the elements of are larger than

Let be a nonstandard model of Here, is treated as a relation, and in we may have placed whatever functions and relations we may have need to reference in what follows; moreover, we assume that in our language we have a constant symbol for each real number. (Of course, this means that we are lifting the restriction that languages are countable.) To ease notation, let’s write for The convention is that we identify actual reals in with their copies in so we write rather than etc.

Show that is a nonstandard model of the theory of problem 1. (In particular, check that the indicated restrictions of and have range contained in )

A (nonstandard) real is finite iff there is some (finite) natural number such that Otherwise, it is infinite. A (nonstandard) real is infinitesimal iff but for all positive (finite) natural numbers one has that We write to mean that either is infinitesimal, or else it is Show that infinite and infinitesimal numbers exist. The monad of a real is the set of all such that which we may also write as and say that and are infinitesimally close. Show that the relation is an equivalence relation. Show that if a monad contains an actual real number, then this number is unique. Show that this is the case precisely if it is the monad of a finite number. In this case, write to indicate that the (actual) real is in the monad of We also say that is the standard part of

Show that a function is continuous at a real iff for all infinitesimal numbers

Suppose that is continuous on the closed interval Argue as follows to show that attains its maximum: For each positive integer there is some integer with such that Conclude that the same holds if is some infinite natural number, i.e., there is some (perhaps infinite) “natural number” with such that Let and argue that the maximum of is attained at

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Given a language and an -structure a set is definable iff there is a formula with (distinct) free variables and there are elements such that, letting be the set of assignments such that for then for all with

In human: is definable iff it is the set of elements of that satisfy some formula. We allow said formula to use parameters, i.e., to refer to some fixed elements of

Type ‘latex’ immediately following the dollar sign, leave a space, and then the math text as you’d do in latex usually. See this announcement for more info.

The wordpress people tweak with the way latex is compiled every now and then, so sometimes strange errors that were not there before appear; but it works pretty decently, and it is getting better. (There seem to be a few silly things still: you want to write {} right before a [ if this is the first symbol in a math display, for example.)

Luca Trevisan devised a nice program, LaTeX2WP, to make the use of in WordPress pleasant rather than traumatic; I use it whenever I have a long post.

Now my concern is: If 0 is not an infinitesimal, then is reflexive. Namely, if then . That is, for all positive . But . So, cannot be infinitesimal. What am I missing here?

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

I give an example (perhaps the best-known example) below, but let me first discuss equiconsistency rather than straight equivalence. Usually an equiconsistency is really the sort of result you are after anyway: You want to establish that certain statements in the universe where choice holds correspond to determinacy, which implies the failure of choice. The […]

The other answers have correctly identified the issue. Let me highlight the difficulty: it is relatively consistent with the axioms of set theory except for the axiom of choice that there are infinite sets which do not contain a copy of the natural numbers (that is, there are infinite sets $X$ such that there is no injection $f\!:\mathbb N\to X$). This means […]

This is $\aleph_\omega^{\aleph_0}$. First of all, this cardinal is an obvious upper bound. Second, if $A\subseteq\omega$ is infinite, $\prod_{i\in A}\aleph_i$ is clearly at least $\aleph_\omega$. The result follows, by splitting $\omega$ into countably many infinite sets. In general, the rules governing infinite products and exponentials are far from being w […]

If $\lambda$ and $\kappa$ are cardinals, $\lambda^\kappa$ represents the cardinality of the set of functions $f\!:A\to B$ where $A,B$ are fixed sets of cardinality $\kappa,\lambda$ respectively. (One needs to check this is independent of which specific sets $A,B$ we pick, of course.) At least for finite numbers, this is something you may have encountered in […]

R. Solovay proved that the provably $\mathbf\Delta^1_2$ sets are Lebesgue measurable (and have the property of Baire). A set $A$ is provably $\mathbf\Delta^1_2$ iff there is a real $a$, a $\Sigma^1_2$ formula $\phi(x,y)$ and a $\Pi^1_2$ formula $\psi(x,y)$ such that $$A=\{t\mid \phi(t,a)\}=\{t\mid\psi(t,a)\},$$ and $\mathsf{ZFC}$ proves that $\phi$ and $\psi […]

What was the precise definition of “definable” again. I can’t find it in the book anywhere.

Given a language and an -structure a set is

definableiff there is a formula with (distinct) free variables and there are elements such that, letting be the set of assignments such that for then for all withIn human: is definable iff it is the set of elements of that satisfy some formula. We allow said formula to use parameters, i.e., to refer to some fixed elements of

Thanks.

Is 0 considered an infinitesimal? By the definition above, 0 would be, but I always thought it was otherwise.

Ah, you are right! I’ve modified the text accordingly.

Making infinitesimals different from 0 now forces us to change slightly the definition of so I’ve done that as well.

Thank you. It’s clear now.

Also, how do you get LaTex to work on your blog? I noticed that you got the approximation symbol to show, but when I tried approx it didn’t work

Type ‘latex’ immediately following the dollar sign, leave a space, and then the math text as you’d do in latex usually. See this announcement for more info.

The wordpress people tweak with the way latex is compiled every now and then, so sometimes strange errors that were not there before appear; but it works pretty decently, and it is getting better. (There seem to be a few silly things still: you want to write {} right before a [ if this is the first symbol in a math display, for example.)

Luca Trevisan devised a nice program, LaTeX2WP, to make the use of in WordPress pleasant rather than traumatic; I use it whenever I have a long post.

Now my concern is: If 0 is not an infinitesimal, then is reflexive. Namely, if then . That is, for all positive . But . So, cannot be infinitesimal. What am I missing here?

[Addressed by the revised definition. -A.]