3
votes
0answers
40 views

Proof on Dyadic Trees [Smullyan: First-Order Logic, chapter 1, section 0]

I'm having difficult with a proof from Smullyan's First-Order Logic, Chapter 1 Section 0 (Reprint, Dover 1968, p. 4): Prove: In a dyadic tree, define x to be to the left of y if there is a ...
2
votes
2answers
110 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
3
votes
1answer
68 views

isomorphism between divisible, totally ordered, abelian groups

Let $G$, $H$ be divisible, abelian, linearly ordered groups, whose cardinalities are equal and satisfy $\mu := |G|=|H|>\aleph_{0}$. These are supposed to be (order!) isomorphic. And just about ...
4
votes
1answer
82 views

Infinite linear order with endpoints which is non-dense

In the process of answering questions about normal models, I had to prove the following: Any normal model of $\chi$ is a non-dense linear order with a least and greatest element. The next question ...
4
votes
2answers
96 views

Is $a \le b$ a true statement if $a < b$? [duplicate]

My question is: Is $a \le b$ true if $a < b$? For instance: Is $3 \le 4$ a true statement? I think yes, because $a \le b$ is defined as $a < b\vee a = b$ and this should be true, even if $a = ...
-3
votes
1answer
206 views

What's the difference between relations, functions, and operations?

My understanding is vague: operations is a subclass of functions, functions is a subclass of relations. operations and functions can form terms but not formulas, relations can form formulas but not ...
0
votes
1answer
42 views

Completeness of posets

I think this is rather a simple question in order theory, but if someone could explain it step by step that would be really useful. If we have an arbitrary set, then let's denote the set of all finite ...
0
votes
2answers
57 views

Proving that Order for N is Anti-symmetric

I'm having trouble deriving the following fact from the basic properties of $+_{\mathbb{N}}$ and the definition: Definition. $\forall n, m \in \mathbb{N}: (n \geq m) \leftrightarrow (\exists a \in ...
3
votes
1answer
78 views

Does model theory extend to partial functions?

I have been reading a bit about effect algebras and d-posets recently, sets $M$ on which you have a single partial binary operation (partial here meaning partially defined, i.e. the domain of this ...
2
votes
1answer
64 views

A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
2
votes
1answer
71 views

Equality and order in sets

Just started Baby Rudin and got struck in this. While defining order in sets, $<$ was introduced as a relation and for a set to be ordered the condition was: for all $x,y$ belonging to an ordered ...
0
votes
1answer
121 views

Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
8
votes
1answer
206 views

Is this a characterization of well-orders?

While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I ...
3
votes
3answers
118 views

Set $A$ has the cardinality of $\aleph_0$. Prove some properties of partial order $ \langle \mathcal P (A), \subseteq \rangle$.

I try to solve the following task: Set $A$ has the cardinality of $\aleph_0$. The truth is that in the partial order $ \langle \mathcal P (A), \subseteq \rangle$: (answer true or false) a) ...
4
votes
4answers
123 views

Show that Total Orders does not have the finite model property

I am not sure whether my answer to this problem is correct. I would be grateful if anyone could correct my mistakes or help me to find the correct solutions. The problem: Show that Total Orders ...
5
votes
1answer
73 views

$T\vDash\psi$ equivalences

$T\vDash\psi$ means $T$ satifies $\psi$ from Tarski's definition of truth, it simply means that the sentence $\psi$ is valid in $M$. I call a sentence $\psi $ universally if it is valid in every ...
5
votes
1answer
124 views

Semi-formal language - Universe has at least three elements

First of all I would like to construct a semi formal sentence, such that the universum has at least three elements. My attempt: $$\exists x\exists y\exists z (x\not=y\wedge y\not=z\wedge x\not=z)$$ ...
4
votes
1answer
215 views

Exclude operation symbols in signature

We probably know what a signature is, it contains a set $\sigma_{op}$ (the operation symbols), $\sigma_{rel}$ (the relation symbols) and a function $ar:\sigma_{op}\cup\sigma_{rel}\rightarrow\mathbb ...
4
votes
2answers
66 views

Semi-formal language

I want use semi-formal language to describe the following four points. (1) The group axioms with signature $\{*\}$ (2) The property "linear order" with signature $\{<\}$ The following properties ...
8
votes
2answers
214 views

Ordinal interpretation of Friedman's $n$?

I heard that Kruskal's tree theorem can be turned into a finite form that creates an extremely fast growing function because ordinals could be encoded into trees. On this wiki page it mentions that ...
3
votes
2answers
77 views

Class of structures isomorphic to $(\mathbb{Z},<)$ in infinitary logic $L_{\omega_1 \omega}$

We want to define the class of all structures isomorphic to $(\mathbb{Z},<)$ in the infinitary logic $L_{\omega_1 \omega}$. Therefor we define strict order as usual: $\forall x,y,z ~~ (x<y ...
2
votes
2answers
70 views

Reaching elements with finite applications of the successor relation in well-ordered sets

Given an well-ordered class $(A,\leq)$ we want to proof that every element $a \in A$ can be reached by applying the successor relation on a the smallest element of $A$ or on an limes element of $A$ ...
3
votes
3answers
222 views

Ways Of Ordering A Set

Perhaps this is a strange question. But it made me think and I have no idea how to answer it. Or if it actually makes sense. I've come across "Well-Ordered Sets" a few times now, and it's well known ...
2
votes
0answers
82 views

Ordering of multisets in “Paramodulation based theorem proving”

I'm reading this paper: http://www.lsi.upc.edu/~albert/papers/handbook.ps.gz and I can't understand a part of it. it defines an ordering on multisets (it defines a multiset over $A$ as a function $A ...
1
vote
2answers
159 views

Cofusing partial order “implies”, on logic and that on sets

I feel confused comparing partial order on sets and that on logic. $$x\ge 1 \implies x\ge 0$$ Here we see a smaller set "implies" a bigger set But, if we know three facts, fact1,fact2,fact3, we ...