set theory – What is first-order logic with Dedekind-finite sets of variables?

The usual set up of first-order logic is with an infinite reservoir of variables which we can use in formulas. This is one of the annoying reasons why we need to put $aleph_0$ into the cardinal equations, but it also provides us with the expressive freedom that we need.

It is not hard to see that there is no reason to consider any other cardinal, except $aleph_0$, in this case: since formulas are finite, and proofs are finite, in any kind of proof there will only be finitely many variables. So anything larger than $aleph_0$ is kind of irrelevant. But this assumes that all cardinals are comparable, and the Axiom of Choice makes things kinda nice.

Assuming that the Axiom of Choice fails, and badly, what happens when we substitute the reservoir of variables with some Dedekind-finite set? In particular, a set $A$ whose finite subsets (and finite injective sequences) form a Dedekind-finite set, or even an amorphous set?

Can we prove some interesting results (read: not entirely equal to standard first-order logic) in $sf ZF$ (+ whatever set we needed exists), or at least some consistency results? For example, we don’t need choice to prove that every theory in a finite language has a complete theory extending it, or that it has a model. What happens when we switch to this abominable version of first-order logic?

ho.history overview – Has any open/difficult problem in ordinary mathematics been solved only/mostly by appeal to set theory?

We know that many (if not all) mathematical notions can be reduced to the talk of sets and set-membership. But it nevertheless sounds like a grueling task (if at all possible) to actually get advanced results in many branches of ordinary mathematics if we only work with sets and set-membership relation in our language, or otherwise only rely on set theory. To put it differently: it seems that in order to get results in many branches of mathematics one might not need to be very familiar with set theory at all, let alone being able to translate everything to the language of sets or to heavily rely on set theory.

I’m wondering if there are cases where an open/a difficult problem in other branches of mathematics (e.g., number theory or real analysis) has been solved mostly/only because of the insight that set theory has offered, directly or indirectly (say, through branches that heavily appeal to set theory, such as model theory). Even a historical incident will be helpful: a problem of the sort that was first solved thanks to set theory, but later on more accessible solutions have been found that don’t deal much with sets.

Thank you very much!

Google Analytics Custom Dimension for affiliate tracking is not being set

I’m trying to set up affiliate link tracking on my website but I’m clearly missing a step and don’t know how to get this working.

My intended flow is for someone to enter my site with an “affiliate_id” in the URL (ex: GTM then uses a custom HTML tag to create and store it in a 1st Party Cookie whenever it queries the URL and finds the “affiliate_id”. In my Google Analytics Settings variable, I have a Custom Dimension at index 1 set as my 1st Party Cookie(which is the affiliate_id value). In GA I created a Custom Dimension at index 1 to receive the affiliate_id that we set up in GTM.

– I see all of the variables properly set when using Tag Assistant.
– My website is created through Wix, which is a SPA (Single Page Application).
– I have an extension in Google Chrome to view cookies and my cookie is correctly set up.
– In GA i have “Enable Ecommerce” set to “On” and “Enable Enhanced Ecommerce Reporting” set to “off”
– I see my test transactions being tracked in GA but they don’t have the Affiliate ID custom dimension set

I really hope someone can help me out. Thanks!



The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex

Suppose I wish to find the set of isomorphism classes of $mathbb{Z}/nmathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $mathbb{Z}/nmathbb{Z}$-CW-complex $X$, i.e. $mathrm{Vect}^{1}_{mathbb{Z}/nmathbb{Z}}(X)$, I can write it using equivariant homotopy theory as


Nonequivariantly, $B_{mathbb{Z}/nmathbb{Z}}GL_{1}(mathbb{C})$ is $K(mathbb{Z},2)$ so the above would seem like a Bredon cohomology group $H^2_{mathbb{Z}/nmathbb{Z}}(X;M)$ with $M$ some Mackey functor and I could try to use the universal coefficients spectral sequence to attempt at computing it, however it seems like an awful lot of machinery for one of the simpler cases.

If the action is relatively nice, though it has fixed points, is there a simpler way of finding

If not, which Mackey functor $M$ is the right one?

Thanks for your time

Set of Critical point of a smooth map

Let $F:mathcal{U}subseteqmathbb{R}^{n}rightarrowmathbb{R}^{m}$, $n≤m$, $Finmathcal{C}^{infty}(mathcal{U})$. Let $Crit(F):=lbrace pinmathcal{U}mid dF_{p} is not surjective.rbrace$. Prove that $Crit(F)$ is closed.

I have no idea how to proceed. In general, I would though about an argument with sequences to show that $partial Crit(F)subseteq Crit(F)$ but I don’t know if it’s work. Can someone give me an advice? Thanks before!

set theory – Is having injection to hereditarily size sets equivalent with choice over ZF?

It’s known that ZF proves that for every set $x$ there exists a set $H_x$ of all sets that are hereditarily strictly subnumerous to $x$. [see here]. Now is the following principle equivalent with choice over the rest of axioms of ZF?

For every set $x$, there exists a set $y$ such that: $x$ is subnumerous to $H_y$.

By “subnumerous to” its meant, as usual, possessing an injection towards; and “strictly subnumerous” means, as usual, existence of subnumerousity without existence of a bijection.

price – how to set order total $0 if below $1 after applying discount code or wallet?

I have removed decimal point using this extension. But after applying coupon code order total is in decimal but using this extension decimal value is not showing on frontend.

In this case, If customer using his wallet for the order so he will pay only order total which is showing. So it is creating issue for decimal value which is not showing.

How can I set order total $0(Zero) if decimal value below $1(Like: $0.2, $0.5 etc.)

privacy – What mitigations are there against a timing attack done to find which devices are communicating on a set of devices?

I know the title isn’t good at all but allow me to explain. In this model I have n devices on different networks that are able to communicate with each other. A supervisor is able to see every packet a device sends or receives. How can two devices A and B communicate in such a way that the supervisor knows A and B are talking to someone but doesn’t know they are talking to each other?

I have thought about the mitigation where the communicating devices also send packets to another random device and receives a response. So A sends packets to both B and C; B sends packets to both A and D. Would this work? What are the other methods of mitigation?

turing machines – Can any computably enumerable set be generated by a prefix-free set?

Downey and Hirschfeldt seem to assume that any computably enumerable set of sequences can be generated from some prefix-free set (in the sense that the set of all extensions of the strings in the prefix-free set is equal to the first set). I don’t understand why this would be so.

Specifically, in a proof that a sequence is Martin-Löf random iff is there is no c.e. martingale on the sequence that produces infinite profit, on page 236, D&H assume that for each class $U_n$ that makes up a Martin-Löf test, there is a “prefix-free generator” $R_n$ (which I take to be what I described above, cf. p. 4). D&H’s definition of Martin-Löf test is on 231: the sequence of $U_n$ is merely required to be uniformly c.e. s.t. $mu(U_n)leq 2^{-n}$.

I don’t understand why such a generator must always exist.

For example, let $U_n$ be${00000ldots}$ for all $n$. Then each $U_n$ is null with respect to the uniform measure, so this is a Martin-Löf test. However, any finite sequence of zeros that would include a sequence of all zeros as an extension, would also have extensions such as $01ldots$, $001ldots$, etc., which are not in $U_n$. So there is no generator of $U_n$.

Clearly I am misunderstanding something (or have not noticed some constraint on Martin-Löf tests?).