Crow's Nest 2021-01-21T04:36:17.057Z http://tripedal-crow.github.io/ Raven Hexo The Baltic Seminar Notes \# 1 http://tripedal-crow.github.io/2021/01/21/Baltic_1/ 2021-01-21T04:25:43.000Z 2021-01-21T04:36:17.057Z

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Ralf Schindler - Talk 1 on Logic Summer School of Fudan University, 2020 http://tripedal-crow.github.io/2020/11/24/Talk1/ 2020-11-24T09:29:09.000Z 2020-11-24T09:37:45.589Z Content:

• Stationary sets;
• Forcing revisited;
• Forcing Axioms: $\mathbf{MA}$;
• Proper forcing; semi-proper forcing; stationary set preserved forcing;
• $\mathbf{PFA}$, $\mathbf{SPFA}$, $\mathbf{MM}$.

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<p>Content:</p> <ul> <li>Stationary sets;</li> <li>Forcing revisited;</li> <li>Forcing Axioms: $\mathbf{MA}$;</li> <li>Proper forcing; semi-proper forcing; stationary set preserved forcing;</li> <li>$\mathbf{PFA}$, $\mathbf{SPFA}$, $\mathbf{MM}$.</li> </ul>
Ralf Schindler - Talk 2 on Logic Summer School of Fudan University, 2020 http://tripedal-crow.github.io/2020/11/24/Talk2/ 2020-11-24T09:29:09.000Z 2020-11-24T09:37:38.616Z Content:

• Restate $\mathbf{PFA}$, $\mathbf{SPFA}$, $\mathbf{MM}$ as well as $\mathbf{PFA}^{++}$, $\mathbf{SPFA}^{++}$, $\mathbf{MM}^{++}$;
• A few words on iterated forcing
• Supercompact Cardinals, Laver functions;
• Forcing $\mathbf{SPFA}^{(++)}$
• Weak reflection principle;
• $\mathbf{MM}\Rightarrow2^{\aleph_1} = \aleph_2$.

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<p>Content:</p> <ul> <li>Restate $\mathbf{PFA}$, $\mathbf{SPFA}$, $\mathbf{MM}$ as well as $\mathbf{PFA}^{++}$, $\mathbf{SPFA}^{++}$, $\mathbf{MM}^{++}$;</li> <li>A few words on <em>iterated</em> forcing</li> <li>Supercompact Cardinals, Laver functions;</li> <li>Forcing $\mathbf{SPFA}^{(++)}$</li> <li>Weak reflection principle;</li> <li>$\mathbf{MM}\Rightarrow2^{\aleph_1} = \aleph_2$.</li> </ul>
Draft of My Fine Structure Notes http://tripedal-crow.github.io/2020/10/10/Fine_Structure/ 2020-10-10T08:39:41.000Z 2020-10-20T08:00:14.566Z Notice: This is a Reading note of the Handbook article written by Zeman & Schindler, about basics of fine structure theory. The notation may vary but the enumeration of theorems, lemmas and corollaries are the same.

### Things I have written so far

• Part I:
• Rudimentarily Closed & Definability;
• $S^A_\gamma$ and $<^A_\gamma$;
• Part II:
• $\Sigma_1$-definability of $S^A_\gamma$ and $<^A_\gamma$;
• $\Sigma_1$-Satisfaction and the Skolem function;
• Condensation Lemma;
• Part III:
• Acceptability & Consequences;
• $\Sigma_1$-Projectum.
• Part IV:
• Reducts & Good Parameters;
• Downward & Upward Extension.

### Things I am planning to add in

#### Constructibility

• $\mathcal L_V$ Theory
• $Fml(x)$. Mind the absoluteness.
• Basic Set Theory. $\Sigma_0$ Comprehension.
• Definability & Rudimentary Closed.
• Applications: $\square_\kappa$ holds in $L$;

#### Jensen’s Preprint Book

• Recursion Theory, similar argument like rudimentarily closedness

# Aim: To give a new way of describing the constructible hierarchy by levels and to apply the uniform $\Sigma_1$-definable $\Sigma$-Skolem function in $\Sigma_n$-Skolem functions, which are not uniformly definable.

## Part 1: Rudimentarily Closed & Definability

Rudimentarily Closed: A structure $M = \langle |M|,\in,A\rangle$ that is closed under following function schemas:

• $f(\vec{x}) = \vec x(i)$;
• (Admissibility)$f(\vec x) = \vec x\cap A$;
• $f(\vec x) = {\vec x(i),\vec x(j)}$;
• $f(\vec x) = \vec x(i) - \vec x(j)$;
• $f(\vec x) = \bigcup_{y\in \vec x(0)} g(\langle y,x_2,…,x_n\rangle)$;
• $f(\vec x) = h(\vec g(\vec x))$;

The collection(as a proper class) of all these functions (called the $rud_A$ functions)is defined independent with the structure. A relation is called $rud_A$ if its characteristic function is $rud_A$. $rud_\emptyset$ is simplified as $rud$.

A variety of functions and relations like $\not\in$ and $x’’y$ can be regarded as $rud_A$ or $rud$(See Handbook). The closure of $rud_A$ functions is denoted as $rud_A(M)$. Notice that here $M$ itself cannot be referred as a $\vec x(i)$.

The $J$-Hierarchy is defined very similar as the $L$-Hierarchy, but only use the limit ordinals. $J^A_0 = \emptyset$ and for limit ordinal $\alpha$, $J^A_{\alpha\omega} = \bigcup_{\gamma<\alpha}J^A_{\gamma\omega}$. For successor ordinals, $J^A_{\alpha+\omega} = rud_A(J^A_{\alpha}\cup{J^A_{\alpha}})$. Actually, Lemma 1.4 shows by induction that $L_\alpha = J_\alpha$ if $\alpha$ is a limit ordinal.

One may argue that the collection of $rud_A$ functions could be very complicated due to the function generating schemas(the last two). Thus we introduce the idea of auxillary hierarchy, denoted as $S^A_\alpha$. It is defined via the closure of 15 defined functions $V^2\to V$, and it actually uses all the ordinals.

Fact. $S^A_\alpha = J^A_\alpha (=L_\alpha[A])$ if $\alpha$ is a limit ordinal.

After introducing the preliminaries, we can now discuss the relation between constructibility and $rud_A$ closedness. Remember that $L_\alpha \subseteq \mathcal P(L_\alpha)$ due to its definition.

Lemma 1.4 Let $U$ be a transitive structure with $A\cap V_{rank(U)+\omega}\subset U$. Then $\mathcal P(U)\cap rud_A(U\cup{U}) = \mathcal P(U)\cap \mathbf\Sigma_\omega^{\langle U,\in,A\rangle}$. In particular, $J^A_{\alpha+\omega}\cap \mathcal P(J_\alpha) = def_A(J_\alpha)$.

Remark. As we shall show later, $rud_A$ functions can only add a finite amount of rank to its variables. Thus to restrict $A\cap V_{rank(U)+\omega}\subset U$, we actually let the $rud_A$ functions be $rud_{A\cap U}$ functions.

$$\mathcal{P}(U) \cap \Sigma_{\omega}^{\langle U, \in, A\rangle}=\mathcal{P}(U) \cap \Sigma_{0}^{\langle U \cup{U}, \in, A \cap U\rangle},$$

since now all the quantifiers are bounded. Now we only need to show that
$$\mathcal P(U)\cap rud_A(U\cup{U}) = \mathcal P(U)\cap \mathbf\Sigma_0^{\langle U \cup{U}, \in, A \cap U\rangle}.$$

“$\supseteq$” By induction we can say that every $\Sigma_0$-definable relation would be $rud_A$. Thus for every $x\in\mathcal{P}(U) \cap \Sigma_{0}^{\langle U \cup{U}, \in, A \cap U\rangle}$, $y\in x$ iff $y\in U$ and for some $\vec x\in U^k$, $R(y,\vec x)$ is true for some $rud_A$ relation $R$. To get $U\cap {y:R(y,\vec x)}\in rud_A(U\cup{U})$ we need to use the fact that $f(y,\vec x) := y\cap{z:R(z,\vec x)}$ is $rud_A$ if $R$ is $rud_A$.

“$\subseteq$” Prove by induction that every $rud_A$ function $f$ satisfies the following property: $v_0\in f(v_1,…,v_m)$ is equivalent with a $\Sigma_0$ formula. (Here the assumption in the Remark is required.) Hence, all sets defined like ${y\in U:y \in f(x_1,…,x_n)}$ is in $\Sigma_0^{\langle U\cup{U},\in,A}({x_1,…,x_n})$.

$\square$

Question. What is the maximal/minima ordinal that is/isn’t $\Sigma_0$-definable without parameter? $\omega$ surely satisfies the condition. Moreover, what about the $\Sigma_{n/\omega}$-definable ordinals?

Answer. For $\Sigma_0$-definable ordinals, they are referred as the recursive ordinals. $\omega_1^{CK}$ is the supremum of them, but not recursive itself.

Remark. The core idea of this proof is that $\Sigma_0$-definable relations are also $rud_A$, and the converse is also true. It is quite difficult to see in the direct defintion of $rud_A$, but if one refers to the auxillary hierarchy, then it becomes clear.

Example. Not all $\Sigma_0$ functions (in $ZF$) are rudimentary: $f:x\mapsto \omega$ gives an example of $\Sigma_0$-definable functions which are not rudimentary. The point is, it is easy to verify that every rudimentary function has the so-called finite rank property:

Fact. If $f:V^k\to V$ is $rud$, then there is a $n\in\omega$ such that:
$$f(x_1,…,x_k)<\max_{i\leq k}(rank(x_i))+n.$$
$\square$

## Part 2: Problems arised in $L$-Hierarchy

Being another construction of $L[A]$, the $J$-Hierarchy has many properties that is very similar with the $L$-Hierarchy. Actually many theorems and lemmata are made in comparision with the $L$-Hierarchy.

In the Handbook article, the authors introduced many interesting properties of $J$-Hierarchy. Many similar properties (as well as their proofs) are also included in Devlin’s book Constructibility. Our next goal is to show that in the $J$-Structure, there is a uniformly $\Sigma_1$-definable $\Sigma_1$ Skolem function. We shall remark the usefulness of this statement at the end of this part.

The proofs and explanations are quite tedious, so I just listed all relevant results below.

• “$x=J_\beta^A$” is uniformly $\Sigma_1$-definable over $J_\alpha^A$ where $\beta<\alpha$.
• Similarly, a well-ordering $<^A_\beta$ is uniformly $\Sigma_1$-definable over $J_\alpha^A$ where $\beta<\alpha$.
• The satisfaction $\vDash^{\Sigma_0}_M$ is $\Delta_1$-definable. Thus $\vDash^{\Sigma_1}_M$ is $\Sigma_1$-definable.
• (Condensation) Any transitive structure that is $\Sigma_1$-embeddable into some $J$-structure is itself a $J$-structure.
• If $\alpha$ is closed under Godel’s Pairing funcion (in particular, $\alpha$ is a cardinal), then there is a $\Sigma_1$-definable function $f:\alpha\to J_\alpha^A$ without parameter. For arbitrary $\alpha$ the function requires parameters.

Another usage of the auxillary hierarchy is the $\Sigma_1$-definition of itself and a well-ordering:

Lemma. Both $\langle S^A_\gamma:\gamma<\alpha\rangle$ and $\langle <^A_\gamma:\gamma<\alpha\rangle$ are $\Sigma_1^{J^A_\alpha}$ uniformely definable(the definition does not depend on $\alpha$).

Remark. It is easy to show by induction which is discussed in Jensen’s notes and the Handbook article in detail. We only need to notice that the $\exists$ quantifier is used to declare the level.

Definition. Denote $#(\phi(a))$ as the Godel number of the $\Sigma_0$-sentence $\phi(a)$, with $a$ being the free variable. We call the function
$$f:\omega\to M, #(\phi(a))\mapsto x$$
the $\Sigma_1$ Skolem function iff $M\vDash \phi(x)$. Thus, $M$ satisfies the $\Sigma_1$-sentence $\exists a(\phi(a))$.

Remark. The Skolem function, in short, is a function that witnesses the truthfulness for the $\exists$ quantifier. As Brouwer would agree, a existence sentence couldn’t be viewed as proved if the corresponding elements are not constructed.

In Proposition 1.12, the defined function $f$ maps $i<n$, the Godel number of some $\Sigma_0$-formula(with some free variables), to the collection of all $v(i)$-tuple which satisfies the formula. Notice that $f$ is only a finite initial segment.

Proof. The proof uses finite induction and for each regular step, uses the Lemma 1.1. That is, the ordered pair $\langle n,T\rangle$ we want to add is definable.

We let $\Theta(f,N,n)$ be the $\Sigma_0$-formula which says that “$N$ is transitive, $f = f^N_n$ is defined like above”. Thus the predicate $\vDash^{\Sigma_0}_M$ would be $\Delta_1$, and thus absolute.

Thus for the $\Sigma_1$-definable $\Sigma_1$ Skolem function $h_M$, we only need to “pick out” some elements that is $\Sigma_1$-definable in $f(i)$. The strategy is to pick the $<^A_\beta$-minimal element in $S^A_\beta$, which we proved erlier that these two concepts are both $\Sigma_1$.

Question. I think the reason why he only uses the “first component” is to avoid further tedious problem of definable well-ordering in $n$-tuples of $S^A_\beta$. However I am not quite sure why defining $\Sigma_2$-Skolem function will need $\Sigma_3$ definition.

As an application, we shall now prove the condensation lemma for $J$-Structures.

Theorem. Any transitive structure that is $\Sigma_1$-embeddable into some $J$-structure is itself a $J$-structure.

Proof. Let $\pi:\bar M\xrightarrow[\Sigma_1]{}M$ be such an emebdding, together with $M=\langle J^A_\alpha,\in,B\rangle$. Let $\bar A=\pi^{-1}A$, $\bar B=\pi^{-1}B$ and $\bar \alpha = \bar M\cap Ord$. Now we check $\bar M = \langle J^{\bar A}{\bar \alpha},\in,\bar B\rangle$. Clearly, by the $\Sigma_1$-definability of $S^A_\alpha$, for all $\beta<\bar\alpha$, $S^A{\beta}\in \bar M$, thus $\bar M=J^{\bar A}_{\bar \alpha}$. For the other direction, only to notice that every $x\in\bar M$ is mapped to some $S^A_\beta$ for $\beta<\alpha$.
$\square$

Question. What is the usage of $B$ here?? Is $B$ really affecting the structure of $M$? I don’t think so.

Remark. The below “Convention” is definitely misleading. I think $h_M[X]$ would be much better.

Lemma. If $\alpha$ is closed under Godel paring function, then tere is a surjection $g:\alpha\to J^A_\alpha$ which is $\Sigma^M_1$. For arbitrary ordinal, this function would be $\mathbf\Sigma^M_1$.

Remark. I am not planning to include this proof right now, since we neither introduce the notion of Godel pairing function, nor it has anymore use for us at all. Anyhow, this lemma is interesting and useful.

## Part 3: Acceptability and the $\Sigma_1$ Projectum

In this part we start to construct some of the basic analysis of the $J$-Hierarchy. It turns out that under suitable assumption, $J^A_\rho = H^M_\rho$ and satisfies $ZFC^-$. For the $\Sigma_1$-Projectum, we shall talk about some of the settings for our next note, about upward & downward extension of embeddings.

Definition. A $J$-Structure $M = \langle J^A_\alpha,B\rangle$ is acceptable iff:

• Whenever $\xi<\alpha$ is a limit ordinal and $\tau<\xi$, if $\mathcal P(\tau)\cap J^A_{\xi+\omega}\not\subseteq J^A_\xi$, then there is a surjectve $f: \tau\to\xi$ in $J^A_{\xi+\omega}$(or, $J^A_{\xi+\omega}\vDash |\tau|\geq \xi$.)

Question. The authors wrote that the acceptability can be viewed as a strong version of $GCH$. What does it mean? According to the fact that $L[A]$(or $L_\alpha[A]$) are not generally satisfying $GCH$, $A$ has to satisfy more properties.

Acceptability is a $Q$-property with respect to the following definition.

Definition. $Qv$(called “there are cofinally many $v$”) is the abbriviation of $\forall u\exists v\supset u$. If $\Phi$ is a (fixed) $Q$-sentence, we say a structure $M$ satisfy the $Q$-property defined by $\Phi$ iff $M\vDash \Phi$.

By a cofinal map $\sigma:U\to V$ we declare that $\sigma$ possess the following property:
$$\forall y\in V\exists x\in U(y\subseteq \sigma(x)).$$
Clearly, $\Sigma_1$-elementary embedding is $Q$-preserving downwards. Moreover, if the map is also cofinal, then it is also $Q$-preserving upwards. Thus,

Corollary 1.22 Acceptability is downward absolute with respect to $\Sigma_1$-elementary embeddings. Furthermore, it is (upward) absolute with respect to $Q$-elementary embeddings.

Acceptable structures canbe more well-behaved. For example, Lemma 1.23 says that acceptable structures add no new subsets of sets that are already possessed by lower level with cardinal height.

Proof. While using Lemma 1.17, note that every infinite cardinal is closed under Godel Pairing Function. Consider the surjection $g:\tau\to u$ and the preimage of some $a\subset u$. Towards a contradiction, let the preimage not in $J^A_\rho$. By definition of acceptability, suppose the preimage(a set of ordinals $<\tau$ and $\Sigma_1$-definable) is contained in some $J^A_{\xi+\omega}-J^A_{\xi}$, we have a surjection $f:\tau\to \xi$ in $J^A_{\xi+\omega}$. Thus $\tau\geq\xi\geq\rho$, which is impossible since $\tau<\rho$. Thus the preimage is in $J^A_{\rho}$ and $a\in J^A_{\rho}$. $\square$

Corollary. $|J^A_\rho| = H^M_{\rho}$ provides that $J^A_\rho$ is acceptable and $\rho$ is a cardinal.

Remark. The cardinal property make sure that its size is preserved in lower level.(The comparision of size in lower level is “true”.)

Question. So let us now consider what makes an acceptable $J$-Structure: the set $A$ used to constrct the hierarchy and some ordinal $\alpha$. Is Corollary 1.22 proves that all of the $J$-structure constructed from $A$ is acceptable? Should we refer to the Condensation Lemma?

Lemma 1.24 Let $\langle M,\in, A\rangle$ be an acceptable $J$-structure. Then for any $\rho,\alpha\in Ord^M$, if $\alpha\subset J^A_\rho$ and $Card(a)<\rho$ in $M$, then $\rho$ being an infinite successor cardinal in $M$ gives $\alpha\in J^A_\rho$.

Proof.
$\square$

The proof of Lemma 1.23 gives us the intuition to find the $\Sigma_1$ Projectum in some acceptable $J$-Structure.

Definition. The $\Sigma_1$ Projectum $\rho(M)$ is defined as the least ordinal which satisfies
$$\mathcal P(\rho(M))\cap\mathbf \Sigma_1^M\not\subseteq M.$$

Remark. Equivalently, $\rho = \rho(M)$ is the $\Sigma_1$-Projectum iff there exists a $\mathbf{\Sigma}_1$-definable function $f$ in $M = J^A_\alpha$ such that
$$f’’J^A_{\rho} = J^A_\alpha.$$

In our proof above, we actually shows that $\Sigma_1$-Projectum can only be a $\Sigma_1$-cardinal, which means there are no $\Sigma_1$-definable surjection $f:\tau\to \rho(M)$ for some $\tau<\rho(M)$. In particular, $\rho(M)$ would be a cardinal if $\rho(M)\in M$.

Proof.
$\square$

Remark. The above theorem shows that the two definitions are indeed equivalent.(…)

## Part 4: Downward & Upward Extensions

Recall that our aim is to use the uniformly definable $\Sigma_1$-Skolem function to analysis $\Sigma_n$ properties. In this part we shall introduce some…

Definition. Let $M = \langle J^B_\alpha,D\rangle$ be an acceptable $J$-structure, $\rho = \rho(M)$ is the $\Sigma_1$ Projectum and $p\in M$. Then
$$A_{M}^{p}=\left{\langle i, x\rangle \in \omega \times J_{\rho}^{A} \mid M \vDash \varphi_{i}(x, p)\right}$$

is called the standard code determined by $p$. The strucure $M^p = \langle J^B_\rho,A^p_M\rangle$ is called the reduct determined by $p$. Moreover, the set
$$P_M = {p \in[\rho(M), M \cap \mathrm{On})^{<\omega}:\exists B\in\Sigma^M_1({p})[B\cap \rho\not\in M]}$$

is called the set of good parameters; the set
$$R_M = {r:p \in[\rho(M), M \cap \mathrm{On})^{<\omega}:h_M(\rho\cup{r}) = M}$$

is called the set of very good parameters.

It is easy to see that $R_M\subset P_M$, and both of them are non-empty. Now we would like to show that $\Sigma_0$-elementary embeddings between reducts defined by very good parameters can be extended to the whole structure, which is also $\Sigma_1$-preserving.

Lemma 3.1 Let $\pi: \bar M^{\bar p}\xrightarrow[\Sigma_0]{}M^p$, where $\bar p\in R_{\bar M}$. Then there exists a unique extension $\tilde \pi:\bar M\xrightarrow[\Sigma_0]{} M$ with $\tilde\pi(\bar p) = p$. Moreover, $\tilde \pi$ is $\Sigma_1$-preserving.

Proof.
$\square$

Remark. The above lemma actually shows that every $\Sigma_n$-elementary embedding between reducts can be extended to a unique $\Sigma_{n+1}$-elementary embedding between the original structures, given that $p\in R_M$ also.

Proof.
$\square$

Now we can give a generalized Condensation Lemma for reducts defined by a very good parameter.

Lemma 3.3. Let $\pi:N\xrightarrow[\Sigma_0]{} M^p$, where $N$ is a $J$-strcture and $p\in R_M$. Then there exists a unique $\bar M$ and $\bar p\in R_{\bar M}$ such that $N = \bar M^{\bar p}$.

Proof.
$\square$

The above lemma, together with Lemma 3.1 and 3.2, is called the “downward extension of embeddings”. To give a dual lemma, we first notice that such a extension of the codomain may not exist. Thus, we need to strengthen the power of the embedding.

Definition. A $\Sigma_1$-elementary embedding $\pi$ is called strong iff the well-foundedness of rudimentary functions with the same definition is preserved under $\pi$.

Lemma 4.1. Let $\pi:\bar M^{\bar p}\to N$ is strong, where $N$ is acceptable and $\bar p\in R_{\bar M}$. Then there are unique $M$ and $p\in R_M$ such that $N = M^p$. Moreover, the extended embedding $\tilde \pi:\bar M\xrightarrow[\Sigma_1]{} M$ satisfies $\tilde \pi(\bar p)=p$.

Proof.
$\square$

## Reference

 Handbook article of Zeman & Schindler;

 Devlin’s book Constructibility;

 Jensen’s unpublished notes Manuscript on fine structure, inner model theory, and the core model below one Woodin cardinal. It is on his personal website.

 Sy D. Friedman’s book Fine Structure and Class Forcing;

 Jensen’s original paper of $J$-Hierarchy The fine structure of the constructible hierarchy;

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<p>Notice: This is a Reading note of the Handbook article written by Zeman &amp; Schindler, about basics of fine structure theory. The notation may vary but the enumeration of theorems, lemmas and corollaries are the same.</p> <h3 id="Things-I-have-written-so-far"><a href="#Things-I-have-written-so-far" class="headerlink" title="Things I have written so far"></a>Things I have written so far</h3><ul> <li>Part I:<ul> <li>Rudimentarily Closed &amp; Definability;</li> <li>$S^A_\gamma$ and $&lt;^A_\gamma$;</li> </ul> </li> <li>Part II:<ul> <li>$\Sigma_1$-definability of $S^A_\gamma$ and $&lt;^A_\gamma$;</li> <li>$\Sigma_1$-Satisfaction and the Skolem function;</li> <li>Condensation Lemma;</li> </ul> </li> <li>Part III:<ul> <li>Acceptability &amp; Consequences;</li> <li>$\Sigma_1$-Projectum.</li> </ul> </li> <li>Part IV:<ul> <li>Reducts &amp; Good Parameters;</li> <li>Downward &amp; Upward Extension.</li> </ul> </li> </ul>
Course Notes of Axiomatic Set Theory http://tripedal-crow.github.io/2020/09/24/Course-Notes/ 2020-09-24T07:36:13.000Z 2020-09-24T08:11:18.000Z This is a collective course note taken in Prof. Simon Thomas’ course Axiomatic Set Theory of Rutgers University, which is held on Spring semester, 2019. The main topic of this course is forcing, forcing axioms such as $\mathbf{MA}$, Open Coloring Axiom, Axiom A of Baumgartner and the Proper Forcing Axiom of Shelah. Also, this course discussed the relation among themselves and basic independent statements like $\mathbf{CH}$. The main reference would be Kunen’s book and Jech’s book. If there is any mistakes or comments, please feel free to contact me.

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<p>This is a collective course note taken in Prof. Simon Thomas’ course <em>Axiomatic Set Theory</em> of Rutgers University, which is held on Spring semester, 2019. The main topic of this course is forcing, forcing axioms such as $\mathbf{MA}$, Open Coloring Axiom, Axiom A of Baumgartner and the Proper Forcing Axiom of Shelah. Also, this course discussed the relation among themselves and basic independent statements like $\mathbf{CH}$. The main reference would be Kunen’s book and Jech’s book. If there is any mistakes or comments, please feel free to contact me.</p>

1. 经验论的数学哲学：不存在关于现实的先天知识。数学无关乎现实世界，数学命题是分析命题。在现实世界的（“出乎意料的”）有效性是纯粹偶然的。对于更一般的科学来说，所有科学命题都是“大概的假设”。
2. 数学是有意义的，但数学命题的意义并非在于其在物理世界的映照，而是在于论证分析命题在我们的世界（一个低智商人群组成的世界）是合理的。“一个具有无限智力的人对逻辑和数学毫无兴趣。”
3. 批驳康德主义唯理论：“5+7=12”不是综合命题，是分析命题，错误的根源在于康德使用了非逻辑的论据；几何学是分析的且无关乎人类空间直觉的，因为几何学也不描述物理世界：康德受到了时代的限制。（康德的这个错误常常使得现代数学家忍俊不禁。）

“密尔坚持认为逻辑和数学的真理不是必然的或确定的。他坚持说这些命题是依据极多次数的事例所做的归纳概括。”

“因为如前所见，他认为命题‘7+5=12’是综合命题的理由是‘7+5’的主观内涵不包含‘12’的主观内涵；而他认为‘所有物体都是延伸的’是一个分析命题的理由仅仅是根据矛盾律。”

“我想，如果我们说一个命题，当它的有效性完全依靠他所含有的符号定义时是分析的，当它的有效性由经验事实所确定时是综合的，那么……”^1

“于是我们看到，存在着一种意义，在这一以上，分析命题确实给我们提供新知识。”

“一个分析命题的有效性绝不依靠于它能从其他分析命题推出，这一事实使我们有理由漠视数学命题是否能像罗素所设想的那样归约为形式逻辑命题这个问题。”

“正像维特根斯坦所说的，我们认为，不能想象世界违反逻辑规律的理由，只不过是我们不能说出一个非逻辑的世界应是什么样的。”

“因为一个选择的好的定义将引起我们对分析真理的主义，否则这种真理会从我们面前溜走。而构造合用的和富有成果的定义很可以被看成一种创造性的活动。”

^1: 我在写的时候突然意识到这种定义方法和康德的存在一定相似性。我的根据在于，我们如何分辨形式证明和经验证明能力的界限呢？打个比方就是，一个数学规范较差的人会尝试实验来判断一个数学命题的真假，那么我们显然不能因此说该判断是正确的。因此，这个命题其实是说的关于一个命题类的性质。

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<p>《先天性》，节选自阿尔弗雷德·朱尔斯·艾耶尔《语言，真理和逻辑》<br>收录在《数学哲学》第三编：数学真理</p> <p>我感觉这篇文章的主要思想还是和我很贴近的。主要在以下方面：</p> <ol> <li>经验论的数学哲学：不存在关于现实的先天知识。数学无关乎现实世界，数学命题是分析命题。在现实世界的（“出乎意料的”）有效性是纯粹偶然的。对于更一般的科学来说，所有科学命题都是“大概的假设”。</li> <li>数学是有意义的，但数学命题的意义并非在于其在物理世界的映照，而是在于论证分析命题在我们的世界（一个低智商人群组成的世界）是合理的。“一个具有无限智力的人对逻辑和数学毫无兴趣。”</li> <li>批驳康德主义唯理论：“5+7=12”不是综合命题，是分析命题，错误的根源在于康德使用了非逻辑的论据；几何学是分析的且无关乎人类空间直觉的，因为几何学也不描述物理世界：康德受到了时代的限制。（康德的这个错误常常使得现代数学家忍俊不禁。）</li> </ol>
Ralf Schindler - Talk 5 on Logic Summer School of Fudan University, 2020 http://tripedal-crow.github.io/2020/09/23/Talk5/ 2020-09-23T08:37:02.000Z 2020-09-23T08:51:08.000Z Content:

• Finish the last theorem of the last lecture: Force by a stationary set preserving forcing:
$$(M;\in,I)\xrightarrow[\text{of length } \omega_1]{\text{generic iteration}}(H_{\omega_2}^V;\in,\mathbf{NS}_{\omega_1}^V),$$

where $M$ is a generically iterable countable transitive structure.

• $\Bbb P_{\max}$ forcing and analysis of $L(\Bbb R)^{\Bbb P_{\max}}$;

• $(\ast)$ and: $\mathbf{MM}^{++}\implies(\ast)$.

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<p>Content:</p> <ul> <li><p>Finish the last theorem of the last lecture: Force by a stationary set preserving forcing:<br> $$(M;\in,I)\xrightarrow[\text{of length } \omega_1]{\text{generic iteration}}(H_{\omega_2}^V;\in,\mathbf{NS}_{\omega_1}^V),$$</p> <p> where $M$ is a generically iterable countable transitive structure.</p> </li> <li><p>$\Bbb P_{\max}$ forcing and analysis of $L(\Bbb R)^{\Bbb P_{\max}}$;</p> </li> <li><p>$(\ast)$ and: $\mathbf{MM}^{++}\implies(\ast)$.</p> </li> </ul>
Ralf Schindler - Talk 4 on Logic Summer School of Fudan University, 2020 http://tripedal-crow.github.io/2020/09/23/Talk4/ 2020-09-23T08:37:01.000Z 2020-09-23T08:46:20.000Z Content:

• Show a characterization of precitiousness;
• $V$ is generically iterable with respect to precitious ideals;
• Discussion of effecitive counterexamples to $\mathbf{CH}$.
• Illustrations of Admissible Club Guessing(ACG)$\implies \mathfrak{u}_2 = \omega_2$.
• Prove ACG follows from $\mathbf{MM}$.

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<p>Content:</p> <ul> <li>Show a characterization of precitiousness;</li> <li>$V$ is generically iterable with respect to precitious ideals;</li> <li>Discussion of effecitive counterexamples to $\mathbf{CH}$.</li> <li>Illustrations of Admissible Club Guessing(ACG)$\implies \mathfrak{u}_2 = \omega_2$.</li> <li>Prove ACG follows from $\mathbf{MM}$.</li> </ul>
Ralf Schindler - Talk 3 on Logic Summer School of Fudan University, 2020 http://tripedal-crow.github.io/2020/09/12/Talk3/ 2020-09-12T15:55:02.000Z 2020-09-23T08:36:03.000Z Content:

• discuss some aspects of stationary sets;
• $\mathbf{MM}\implies 2^{\aleph_1}=\aleph_2$;
• effective counterexample to $\mathbf{CH}$.

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<p>Content:</p> <ul> <li>discuss some aspects of stationary sets;</li> <li>$\mathbf{MM}\implies 2^{\aleph_1}=\aleph_2$;</li> <li>effective counterexample to $\mathbf{CH}$.</li> </ul>

“理论要用来培养未来指挥官的智力，更准确地说，理论应当促使他们自修，而不是跟着他们一起上战场。这好比高明的老师要做的是启发和促进学生发展智力，而不是一辈子牵着他们走路一样。”

“在人类生活中的许多活动中，就算人们所学的知识已经忘得差不多了，人们在需要时也可以从满是灰尘的书本里去寻找，甚至人们日常所用的知识也可能完全是身外之物。”

“大约在19，20世纪之交，数学——‘可靠性与真理性的典范’——似乎是正统的欧几里得学派的真正最后堡垒。但是，甚至欧几里得式的数学结构也必定存在一些缺点，这些缺点使人感到严重不安。因此，所有基础学派的中心问题是：‘一劳永逸地建立数学方法的确实性。’可是，基础研究却出乎意料地得到结论：作为总体来看，按照欧几里得方式重组数学也许是不可能的；至少最有意义的数学理论，像自然科学理论一样，是拟经验的。欧几里得注意在它的真正堡垒中遭到失败了。”

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<blockquote><p>“理论要用来培养未来指挥官的智力，更准确地说，理论应当促使他们自修，而不是跟着他们一起上战场。这好比高明的老师要做的是启发和促进学生发展智力，而不是一辈子牵着他们走路一样。”</p> <footer><strong>克劳塞维茨</strong><cite>战争论</cite></footer></blockquote> <p>评论：这段话应用范围之广使得《战争论》一书的哲学性显露无疑。我现在怀疑这句话是否蕴含了某种认识论的哲学倾向，但还不好说。<br>我们常说，“理论指导实践”，指导到底指的是什么？那么在这里，理论就是指南针，但却不是藏宝图。理论和人的自身理性天赋必须得到良好结合，真正的思想火花才能迸发出来。理论给我们指明可能的方向，在一片原野上设立路标，但就其本身而言，并不指涉特定的进路。然而，个人精力的（可悲的）有限性往往使得我们一生只能追寻一条特定的进路，而这样的人就被尊称为某理论家。</p>
Forcing Over CH http://tripedal-crow.github.io/2020/09/12/Test/ 2020-09-12T15:10:06.000Z 2020-09-24T08:11:13.000Z Let $\mathbb P = Fn(\omega_2\times\omega,2)$ be the collection of all the finite partial functions from $\omega_2\times \omega$ to $2$. Our strategy is:

• to firstly find a collection of dense sets $D_{\alpha\beta}$, such that a generic filter $G$ can be build upon;
• to secondly prove that any generic filter of $\mathbb P$ preserves cardinals.

Lemma. 1 $D_{\alpha\beta}$ are dense sets, where
$$D_{\alpha\beta} = {p\in\mathbb P\mid \exists n\in\omega(\langle\alpha,n\rangle\in dom(p), \langle\beta,n\rangle\in dom(p),p(\alpha,n)\neq p(\beta,n))}.$$

Thus, for any generic filter $G\subseteq \mathbb P$, $G\cap D_{\alpha\beta}\neq\emptyset$ and thus $\bigcup G$ is a function $\omega_2\times\omega\to 2$.

Given that $\mathbb P$ satisfies c.c.c.$^1$, we have

Lemma. 2 $\mathbb P$ preserves cofinality and thus preserves cardinals.

Proof. Let $\kappa$ be a regular cardinal which witnesses the change of cofinality. Thus, there are $\alpha<\kappa$ and $f:\alpha\to \kappa,f\in V[G]$ which maps $\alpha$ cofinally into $\kappa$. Given that $\mathbb P$ satisfies c.c.c., we can find a $F:\alpha\to P(\kappa)$ in $V$ such that:

• $\forall\xi<\alpha(f(\xi)\in F(\xi))$;
• $\forall\xi<\alpha(|F(\xi)|\leq\omega)^{V2}$.

Thus, let $S = \bigcup_{\xi<\alpha}F(\xi)$, we have that $S$ is unbounded in $\kappa$, and $|S| = |\alpha| = \kappa$, contradicting the regularity of $\kappa$.
$\square$

By the above lemmas, for any generic filter $G\subseteq V$, $\bigcup G$ is a function from $\omega_2\times\omega\to 2$, which results in $\omega_2$ many different $\omega\to 2$ functions. And since $\omega_2$ is preserved in $V[G]$, we may say that $V[G]\vDash \neg CH$.

$^1$ About the c.c.c. property of $\mathbb P$: Suppose $A = {p_\alpha\mid \alpha<\omega_1}$ is an uncountable antichain of $\mathbb P$, then it is safe to assume that ${dom(p_\alpha)}$ is a Delta system. Let $r$ be its root, then we can forwardly assume that all $p_\alpha$ agrees on $r$. Thus all $p_\alpha$ are compatible with each other.

$^2$ See Lemma. 5.5 of Chap. VII of Kunen’s book.

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<p>Let $\mathbb P = Fn(\omega_2\times\omega,2)$ be the collection of all the finite partial functions from $\omega_2\times \omega$ to $2$. Our strategy is:</p> <ul> <li>to firstly find a collection of dense sets $D_{\alpha\beta}$, such that a generic filter $G$ can be build upon;</li> <li>to secondly prove that any generic filter of $\mathbb P$ preserves cardinals.</li> </ul> <p><strong>Lemma. 1</strong> $D_{\alpha\beta}$ are dense sets, where<br>$$D_{\alpha\beta} = {p\in\mathbb P\mid \exists n\in\omega(\langle\alpha,n\rangle\in dom(p), \langle\beta,n\rangle\in dom(p),p(\alpha,n)\neq p(\beta,n))}.$$ </p>