This doesn't exactly count as an unpublished forthcoming paper, but the supposed original proof of Fermat's Last Theorem that was "too large to fit in the margin" should probably be mentioned here.

First, a point of clarification: you're not adding the poset but rather forcing with it, and the poset itself remains unchanged after the forcing. The generic object that you're adding is actually a ...

Theorem (ZFC + "There exists a supercompact cardinal."): There is no largest cardinal. Proof: Let $\kappa$ be a supercompact cardinal, and suppose that there were a largest cardinal $\lambda$. Since ...

This is a great question that every set theorist first learning about the beautiful area of forcing should ask. As Andres described, there are a number of ways to understand forcing, but I want to ...

Let me ask you a question instead. The consistency of PA is known to be independent of PA so we have a model of PA that thinks there is a proof of something contradictory like $0 \neq 0$. Therefore, ...

Main Question: (1) Yes, let $A_j: 2^{\aleph_j} \neq \aleph_{j+1}$ (i.e., GCH does not hold at $\aleph_j$). We can do this by simultaneously forcing (via a countable product of posets adding Cohen ...

For Goedel's first incompleteness theorem, you can appeal to the existence of any computably (recursively) enumerable set $A$ that's not computable (recursive). Specifically, suppose $T$ is an $\...

As Andres implicitly pointed out, we may avoid diagonalization by working with ordinals directly. We can appeal to Hartog's Theorem to show that there is an ordinal $\beta$ that does not inject into $...

I would avoid referring to your theorems by your own name unless they have become completely standardized as such. This is especially true when you submit a paper establishing a supposed new result ...

Here are some examples that might help in understanding. If ZFC is consistent, then it follows we have a set model $M$ of the theory. Consider a nonprincipal ultrafilter $U$ on $\omega$ and let $M^{\...

If you assume that ZFC is consistent, then it follows from Gödel's completeness theorem that there is a set model of ZFC. You can then argue from a Platonistic point of view by taking the viewpoint ...

Theorem If $0^{\sharp}$ does not exist and $\lambda$ is a singular cardinal, then any forcing adding subsets to $\lambda$ necessarily adds subsets to a cardinal below $\lambda$. Proof: Let $\mathbb{P}...

No, if $A$ holds in a forcing extension $V[G]$, it need not be forced by $1$ in general. But this is not what is done. Instead, the argument can proceed as follows: In order to show that $1$ forces ...

I highly recommend teaching arithmetic in different bases and base conversion because it aids in the conceptual understanding of our ordinary base ten arithmetic and representations of numbers. If ...

One key observation is that with $3$ functions, we are free to have one of them assume any rational value at any Natural number. This is not possible when we only have $2$ functions where after we ...

To your original question, it seems worth mentioning the point that your hypothesis implies the following: If $|A| = |A \times 2|$ and there is a surjection from $A$ onto $B$, then $B$ injects into ...

As Stefan mentions, the Cantor diagonal argument is completely general to larger cardinals. The difference merely is that the indices of your list range over infinite ordinals rather than just the ...

Suppose $\kappa$ is $\lambda$-supercompact for some $\lambda \geq \kappa$, and let $j: V \rightarrow M$ be an elementary embedding with critical point $\kappa$ such that $j(\kappa) > \lambda$ and $...

There is a generalization of the compactness theorem to infinitary logics that sounds somewhat close to what you want. Specifically, the compactness theorem tells us that every finitely satisfiable ...

The Turing Machine you describe here can actually be constructed (from a practical standpoint also), but it would be tedious and not of much practical use. First note that the finite set of symbols $\...

There was a recent article in the American Math Monthly, Analysis with Ultrasmall Numbers, that might be of interest. For a summary of its implementation in a high school classroom, see http://maths....

(1) Even in the absence of a large cardinal hypothesis, Łoś's theorem still applies so we still have $V \models \varphi(x_1, \ldots, x_n)$ if and only if $V^{\kappa}/U \models \varphi(c_{x_1}, \ldots, ...

A somewhat analogous question can be asked about large cardinals. Specifically, why should we study large cardinals when (a) ZFC cannot prove their consistency and (b) we don't generally appeal to ...

Let $A = T \cup R$ where $R$ is any set of refutable statements from $T$ (i.e., $T \vdash \lnot \varphi$ for all $\varphi$ in $R$). A simple necessary condition for such a $T^{\prime}$ to exist is ...

First, in order to avoid Amit's concern, we may as well assume countable choice. Note that if AD (axiom of determinacy) holds in $L(\mathbb{R})$, then DC (axiom of dependent choice) will also be true ...

Observation: This question has a reverse mathematics flavor in the following sense. By the Compactness Theorem, to prove CON(ZFC), it is sufficient to prove the infinite theory consisting of all ...

Let me restrict and better formalize this question to ask, "If we have some first-order formula $\varphi(x)$ expressible (with parameters) in the language of set theory, when will $\lnot \exists x \...

In case you aren't already aware of this, Jech uses a slightly different poset for Hechler forcing. Specifically, he fixes a family $\mathcal{F} \subseteq \omega^{\omega}$ and lets conditions be of ...

The answer to your more restrictive question is still yes with a reasonable definition of computable sequence (and I'll use your Busy Beaver example in the proof). Specifically, I will provide you ...

I really like this question, and I think it gets to the heart of the study of large cardinals in set theory. To add to Andres's excellent answer, let me talk a little more about inner models and ...