#paper #forcing #Keisler_logic [Location](<file:///home/ur/Dropbox/ur and saf/keisler/Papers/Combining forcing with Keisler’s logic 1.pdf>) ### Abstract. We prove that there exists a countable family of continuous real functions whose graphs, together with their inverses, cover an uncountable square, i.e. a set of the form X × X, where X ⊆ R is uncountable. This extends Sierpiński’s theorem from 1919, saying that S × S can be covered by countably many graphs of functions and inverses of functions if and only if $|S| \leq ℵ1$ . Using forcing and absoluteness arguments, we also prove the existence of countably many 1-Lipschitz functions on the Cantor set endowed with the standard non-archimedean metric that cover an uncountable square. **Notation:** covered by functions = by the functions and their inverses. ### Earlier theorems 1. **Sierpinski:** every square of cardinality $\aleph_1$ can be covered by countably many functions 2. **Zakrzewski:** if a square is covered by countably many functions then it is universally small 3. **Abraham and Geschke:** every set X ⊆ R of cardinality ℵ1 there is a ccc forcing notion adding countably many continuous func- tions that cover X × X. Consequently, under Martin’s Axiom every ℵ1 -square in the plane is covered by a countable family of continuous functions. 4. **Abraham, Rubin and Shelah:** *The Open Coloring Axiom* implies that for every set $X ⊆ 2^ω$ of size ℵ1 there is a countable family of 1-Lipschitz functions that covers X × X . 5. **Shelah 522:** using Keisler’s completeness, there exists in ZFC a planar $F_σ$ set C such that $S × S ⊆ C$ for some uncountable set S, while $P0 × P1 \nsubseteq C$ whenever $P0 , P1$ are perfect sets. ## Results ### Basic results **Theorem 2.1.** Let X be a topological space containing at least two points and suppose there exists a continuous onto mapping $ϕ : X → X^ω$. Then there exists a countable family of continuous mappings of X into itself such that every maximal square covered by this family is uncountable. **Corollary 2.2.** There exists a family of continuous functions of the Cantor space to itself that covers an uncountable square. There exists a family of continuous functions of the Real line to itself that covers an uncountable square. **Proof sketch** transform $\phi$ to a function $f=\phi\circ\pi_{0}\circ\phi$ which satisfies that the preimage of every sequence is uncountable, and then compose $f$ with the projections (+ identity). The Cantor space is homeomorphic to its $\omega$ power, and continuous functions of the Cantor space can be extended to the real line. ### Lipschitz functions **Theorem 3.1.** There exists a countable family of 1-Lipschitz functions on the Cantor set that covers an uncountable square. ^1887f2 #### Step 1 **Lemma 3.4.** A poset P that forces a family $F = \{f_n : n ∈ ω\}$ of 1-Lipschitz functions on the Cantor set $2^ω$ and an uncountable set $X ⊆ 2^ω$ whose square is covered by $F$. P introduces - $\{f_n\}_{n∈ω}$ continuous functions on the Cantor set - one-to-one function $γ : ω_1 → 2^ω$ such that $X=\gamma '' \omega_1$ - $\varrho: [ω_1]^2 → ω$ a function such that $γ(α) = f_{\varrho(α,β)} (γ(β))$ for every $α < β < ω1$ . In order to make things work: - P is ccc - Need to meet $\omega_1$ many dense sets. #### Step 2 Formalize the existence of the above in $L^{\omega}(Q)$ : - $L^{\omega}$ adds to f.o.l a unary prediacte $N$ for the natural numbers and a constant symbol for every natural number - $C$ - unary predicate which will denote the set $X$ - $P$ - binary function that makes the elements of $C$ into elements of $2^\omega$ by requiring ![[Pasted image 20221110115035.png]] - $C$ is made dense in $2^{\omega}$ by adding all sentences of the form ![[Pasted image 20221110115658.png]] - $D$ - a distance funcion on $C$. - $L$ - unary predicate which will denote the Lipschitz functions - $V$ - a binary function which makes the elements of $L$ into functions, which are Lipschitz by requiring ![[Pasted image 20221110115445.png]] - And of course ![[Pasted image 20221110115757.png]] **Theorem** A set of $L^{\omega}(Q)$ sentences in a countable language is consistent iff it has a *standard* model. - Consistency is absolute between transitive models of ZFC. **Def.** A model for $L^{\omega}(Q)$ is *standard* if the natural constants are intrepreted as the "real" natural numbers, these are the only members of $N$, and $(Qx\varphi)$ holds iff the set of elements satisfying $\varphi$ is really uncountable. **Remark** the ccc is required for the model to be standard.