Syllabus. Evaluation. Logic. Logic 2. Logic Exercises. Solutions to Logic Exercises.

We reviewed some concepts from logic:

• Formal sentences.
• Topic list for first-order logic.
• Predicates.
• Connectives.

We continued reviewing (first-order) logic:

1. First-order languages.
2. First-order structures.
3. Syntax. (Terms, Atomic Formulas, Formulas, Sentences)
4. Semantics. ($\mathbf{A}\models \sigma$)

• Quantifiers.
• Practice with tables!.

We continued reviewing (first-order) logic:

1. Provability (see pages 30 and 32 of NST) versus truth. Consequence relations.
2. The Gödel Completeness Theorem.
3. The Compactness Theorem.

• Meaning.

We started to review Chapter 2 of NST (= Elementary Set Theory).

1. The First- and Second-Order Peano Axioms.
2. The ZFC Axioms (up to Comprehension).

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. The ZFC Axioms. Models.
   • Expanding the language. ($\emptyset, \subseteq, \subset, \cap, \bigcap, \cup, \bigcup$, set braces)
   • Sets versus classes.
   • Russell's Paradox.
   • Ordered pairs and relations.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. Recursion.
2. The natural numbers object in a model of ZFC. (Definition + order + arithmetic handout + arithmetic + hints handout + arithmetic slides.)

Four student presentations.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. $\langle \mathbb{N}; 0, S(x), x+y, xy, x^y, <\rangle$.
   • If $n\in\mathbb{N}$ and $k\in n$, then $k\in \mathbb{N}$. Hence each $n\in\mathbb{N}$ is the set of its predecessors.
   • $\mathbb{N}$ is a transitive set of transitive sets.
   • $\langle \mathbb{N}; <\rangle$ is a well-ordered set.
2. $\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}\to \mathbb{C}$.
3. Ordinal Numbers.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. Ordinal Numbers.
• Definition.
• Exercises at seats from the Ordinals Numbers handout!
• A 2-page fragment from NST.
• ON is closed under class-size intersections and set-size unions.
• Burali-Forti Paradox.
• ON is well-ordered in the class sense, and each $\alpha\in$ ON is the set of its $\in$-predecessors.
2. Some questions that arose in the previous meeting (Sep 12).
• Q: How many $4$-element sets are transitive?
  • A: There are 9. First show that any nonempty transitive set contains the element $\emptyset = 0$. Then show that any transitive set with more than one element contains $\{0\} = 1$. Next show that any transitive set with more than two elements contains either the element $\{0,1\} = 2$ or the element $\{1\}$. Next show that a transitive set of four elements has the form $\{0,1,2,X\}$ or $\{0,1,\{1\}, X\}$ for some $X\in \mathcal{P}(\{0,1,2,\{1\}\})$. By examining the possibilities, one finds that there are four transitive sets of the form $\{0,1,2,X\}$ with $2\in X$, four more of the form $\{0,1,\{1\}, X\}$ with $\{1\}\in X$, and one final transitive set of the form $\{0,1,2,\{1\}\}$. The complete list is:
    
    • $\{0,1,2,\{2\}\}$
    • $\{0,1,2,\{0,2\}\}$
    • $\{0,1,2,\{1,2\}\}$
    • $\{0,1,2,\{0,1,2\}\}$
    • $\{0,1,\{1\},\{\{1\}\}\}$
    • $\{0,1,\{1\},\{0,\{1\}\}\}$
    • $\{0,1,\{1\},\{1,\{1\}\}\}$
    • $\{0,1,\{1\},\{0,1,\{1\},\{\{1\}\}\}$
    • $\{0,1,2,\{1\}\}$
• Q: Is the class of limit ordinals a proper class?
  • A: Yes. A class of ordinals is a set if and only if it is bounded above. To see this, assume that $A$ is a set of ordinals. Then $\alpha:=\bigcup A$ is the least upper bound of the ordinals in $A$. This shows that any SET of ordinals is bounded above. Conversely, if $B$ is a class of ordinals that is bounded above, say by $\beta$, then $B$ is a subclass of the set $\{x\in S(\beta)\;|\;x \;\text{is an ordinal}\}$, hence $B$ is a set.

    Finally, the class $L$ of limit ordinals is not bounded above. To see this, assume $\gamma$ is a potential upper bound for $L$. The union of the set $\{\gamma, S(\gamma), SS(\gamma), SSS(\gamma), \ldots\}$ is a limit ordinal strictly above $\gamma$. Summarizing: no $\gamma$ can be an upper bound for the class $L$ of limit ordinals, so $L$ is a proper class.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. Transfinite induction and recursion (I+R).
   • Simple I+R.
   • I+R with parameters.
   • Course-of-values I+R
   • Truncated I+R.
2. Zermelo’s Theorem (= Well-Ordering Theorem).
3. A question that arose in the previous meeting (Sep 15).
• Q: The number of $n$-element transitive sets for $n=0,1,2,3,4$ is $1,1,1,2,4$. This looks like OEIS sequence A001142. Is it?
  • A: No. The number of $n$-element transitive sets is given by OEIS sequence A001192. The two sequences start to differ at $n=5$.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. Cardinal numbers.
   • Definition.
   • Cantor’s Theorem.
   • CBS Theorem.
   • Regular and singular cardinals.
2. Cardinal arithmetic.
   • The Main Theorem of Cardinal Arithmetic can be found on pages 154-156 of NST.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. Cardinals.
   • Definitions.
   • Remarks.
     • Cardinal arithmetic is well-defined.
     • $+$ is commutative, associative, unit $0$.
     • $\ast$ is commutative, associative, unit $1$, absorbing element $0$.
     • $\ast$ distributes over $+$.
     • Exponentiation satisfies the expected rules.
     • Ordinal and cardinal arithmetic agree on $\omega$.
     • Ordinal and cardinal arithmetic disagree on some infinite cardinals. In particular, we do not expect commutativity for ordinal addition or multiplication.
     • The definitions of cardinal arithmetic can be extended to infinite sums and products: e.g., $$\Sigma \kappa_i = \kappa_0+\kappa_1+\kappa_2+\cdots = |\kappa_0\sqcup \kappa_1\sqcup \kappa_2\sqcup\cdots| = |\bigsqcup \kappa_i|.$$
   • Theorem. For all infinite $\kappa$, $\kappa\ast \kappa = \kappa$.
   • Cardinal arithmetic.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. We spent the day discussing equivalent forms of AC (Section 10 of NST), in particular the form that asserts that for every infinite $A$ we have $|A\times A|=|A|$.

Three student presentations.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. One student presentation.
2. We discussed the equivalence of AC and Zorn’s Lemma.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. We discussed the use of Zorn’s Lemma in the ZFC proof that $(\forall A^{\text{inf}})(|A\times A|=|A|)$.
2. We introduced the Hartogs Number ($h(A), H(A)$, or $\aleph(A)$) and the Lindenbaum Number ($L(A)$, or $\aleph^*(A)$) and explained why $\aleph(A)\leq \aleph^*(A)$ for every set $A$.
3. We showed that we can derive AC from ZF+$(\forall A^{\text{inf}})(|A\times A|=|A|)$. An analysis of the argument showed that we actually derived AC from ZF+($(\forall B^{\text{inf}})\;(|B\times \aleph(B)|\leq |B\sqcup \aleph(B)|)$).

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

1. We began by making some comments about cardinal exponentiation:
   • The function $(\kappa,\lambda)\mapsto \kappa^{\lambda}$ is monotone in both variables.
   • It is known that class of triples $(\alpha,\beta,\gamma)$ for which $\aleph_{\alpha}^{\aleph_{\beta}}=\aleph_{\gamma}$ is not determined by the axioms of ZFC. For example, ZFC does not determine whether $\aleph_{0}^{\aleph_{0}}=\aleph_{1}$.
   • We defined the functions א (Aleph), ב (Beth), ג (Gimel).
   • We defined the Continuum Hypothesis (CH) and the Generalized Continuum Hypothesis (GCH).
2. We evaluated $\kappa^{\lambda}$ when (i) $\kappa,\lambda\in\omega$, (ii) $\kappa\in \{0,1\}$, (ii) $\lambda\in \{0,1\}$.
3. We showed that if $2\leq \kappa$, $\lambda$ is infinite, and $\kappa\leq\lambda$, then $\kappa^{\lambda}=2^{\kappa}$. The proof used an instance of the cardinal exponentiation rule $(\kappa^{\lambda})^{\mu} = \kappa^{\lambda\cdot\mu}$, which we discussed.
4. We discussed cofinality for posets in general and for ordinals in particular. We recalled the definition of regular and singular cardinals. Some observations we made about cofinality of infinite cardinals were
   • $\text{cf}(\kappa) \leq \kappa$
   • $\text{cf}(\text{cf}(\kappa)) = \text{cf}(\kappa)$
   • $\text{cf}(\kappa)$ is a regular cardinal.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

Today’s topics:

1. König’s Theorem: If $\lambda_i$ and $\kappa_i$ are cardinals and $\lambda_i<\kappa_i$ for all $i\in I$, then $\sum_{i\in I} \lambda_i < \prod_{i\in I} \kappa_i$.
2. Corollary 1. (Cantor’s Theorem) If $\kappa$ is a cardinal, then $\kappa < 2^{\kappa}$.
3. Corollary 2. (König’s Corollary) $\kappa < \kappa^{\text{cf}(\kappa)}$.
4. Corollary 3. $\aleph_{\alpha} < \text{cf}(2^{\aleph_{\alpha}})$.
5. We began discussing the Main Theorem of Cardinal Arithmentic, Theorem 11.58 of NST. We proved Parts (1), (2), and (3)(a), but not Part (3)(b) yet.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

We completed the discussion of the Main Theorem of Cardinal Arithmetic following these slides.

We continued reviewing Chapter 2 of NST (= Elementary Set Theory).

We discussed the cofinality of cardinals following these slides.

Student presentations.

1. We discussed the questions:
   • What is a model of group theory? What is a model of topology?
   • How do we transition from No Knowledge to Knowledge?
2. We discussed Richard’s Paradox and the need for a distinction between metamathematics and formal mathematics. We briefly recalled an earlier reading assignment from Kunen, namely: Kunen, Chapter 1, Sections 3 and 4.
3. We defined set-theory structures and class models
4. We defined the von Neumann Hierarchy.
5. We explained why every set has a transitive closure, which is a set. As class came to an end, we briefly sketched why every set has a rank.

We started by reviewing some material from the last meeting and filling in some of the proofs. Namely,

1. We recalled the definition of $V_{\alpha}$.
2. We used transfinite induction to prove that $V_{\alpha}$ is a transitive set. We proved that $\beta\leq \alpha$ implies $V_{\beta}\subseteq V_{\alpha}$.
3. We defined the transitive closure of a set $A$, $\text{tr.cl}(A)$, and showed that $\text{tr.cl}(A)$ exists, is a set, and it is the least transitive set containing $A$ as a subset.
4. We defined the ordinal rank of a set, $\text{rank}(A)$, and proved that $\text{rank}(A)$ exists for any set $A$.

1. We reviewed the definition (NST 12.3) and the properties (NST 12.4-8) of rank.
2. We defined the hereditarily finite sets.
3. We discussed the cardinality of $V_{\alpha}$ for infinite $\alpha$.
4. We speculated about the first value of $\alpha$ such that $\langle \mathbb{R}; \cdot, +, -, 0, 1\rangle\in V_{\alpha}$.
5. We discussed why $V$ has no nonidentity automorphisms.
6. We discussed Scott’s Trick for speaking about cardinality in the absence of choice.
7. We briefly discussed the history of the problem of constructing models of set theory. (Fraenkel’s permutation models, Gödel’s L, Cohen’s Forcing, and the Jech-Sochor embedding theorem.)
8. We discussed the Axioms of ZFA following these slides.

We discussed the Fraenkel’s first model following these slides.

We began a discussion of how to show that Fraenkel’s first model satisfies the axioms of ZFA following these slides.

We continued our discussion about how to verify that a transitive class satisfies the axioms of ZF following these slides.

We completed our discussion about how to verify that a transitive class satisfies the axioms of ZF following these slides.

Student presentations.

• We discussed absoluteness and $\Delta_0$-formulas following these slides.

• We discussed the reflection theorem following these slides.

We reviewed the path (= Absoluteness, Reflection Theorem, Mostowski Collapse) to establishing why ZFC+Con(ZFC) proves the existence of a countable, transitive, model of ZFC. We began a discussion of the ideas of Forcing (Chapter 28, NST), with the purpose of explaining how to construct a model of ZFC where CH fails. We introduced the following concepts:

• a forcing order $\langle P; \leq, 1\rangle$.
• (in)compatible elements $p, q\in P$.
• a filter $G\subseteq P$.
• a dense subset $D\subseteq P$.
• a generic filter $G\subseteq P$.

We continued the preceding discussion about Cohen forcing. (Read pages 597-598 of NST.)

We stated the following theorem:

Theorem. (Cohen) Let $M$ bf a c.t.m. of ZFC, let $\kappa$ be an infinite ordinal in $M$, and let $G\subseteq F(\kappa\times \omega,2,\omega)$ be a $\mathbb{P}$-generic filter over $M$. Then

1. $M[G]$ is a c.t.m. of ZFC with the same ordinals as $M$.
2. $M[G]$ has the same cardinals and some cofinalities of limit ordinals as $M$.
3. $\kappa\leq 2^{\omega}$ holds in $M[G]$.

We announced that HW4 is the final HW assignment of the semester.

Today we discussed the $\Delta$-System Lemma (read NST pages 357-359) following these slides. We skipped slide 9 (Counterexamples to Generalizations) to fit the discussion into the time available. You might want to try to verify the details of these counterexamples!

The $\Delta$-system Lemma was proved in ZFC. At the end of class, one student asked how how much of AC is necessary for the proof. I located this paper, which appears to prove that the implication ZFC$\Longrightarrow$(ZF + $\Delta$-System Lemma) is not reversible, but then the paper pins down the strength of the $\Delta$-System Lemma by showing (in Corollary 2.5) that the $\Delta$-System Lemma is equivalent over ZF to the conjunction of the following two consequences of AC:

• A countable union of countable sets is countable.
• Every uncountable collection of countable sets has an uncountable subcollection with a choice function.

We briefly reviewed the definitions of Cohen Forcing, then:

• We proved Lemma 28.1 of NST. (= existence of generic filters.)
• We proved Lemma 28.2 of NST. (= generic filters do not belong to the ground model if the forcing poset is sufficiently branching.)
• We defined $\mathbb{P}$-names and explained their role in the construction of $M[G]$.

We discussed the construction of $M[G]$ following these slides.

We completed these slides on the construction of $M[G]$.

We completed these slides, which introduce the Forcing Theorem.

We completed these slides, which introduce the forcing relations.