Axiom Definition and 147 Threads

This set-theory theorem is very easy to prove: (*) if A≈B & C≈D & A∩C=∅ & B∩D=∅ then A∪C≈B∪D It seems intuitive that if one replaces the strong A∩C=∅ & B∩D=∅ condition by the weaker A∩C≈B∩D the implication (**) if A≈B & C≈D & A∩C≈B∪∩D then A∪C≈BD still holds. (**) does not seem to be much...

Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more? I feel...

prove: The 2nd axiom of mathematical logic 2) $((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))$ By using only the deduction theorem

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focused on Searcoid's treatment of ZFC in Chapter 1: Sets ... I need help in order to fully understand the Axiom of Power Sets and Definition 1.1.1 ... The...

(x)(x)>0 (D) (x+(-x))(x+(-x)) >0 (A4) x^2 + 2(-x)(x) + (-x)^2 >0 (D) x^2 - 2x^2 + (-x)^2 >0 -x^2 + (-x)^2 >0

From Wikipedia entry on the Axiom of Choice: [1] What about a finite set of indistinguishable things (e.g. identical socks)? Do we need to invoke the axiom? [2] Is there any physical consequence of this axiom, i.e. is there any physics experiment where the calculations to predict the result...

Please refer to the screenshot below. Every step is justified with an axiom. Please see the link to the origal document at the bottom. I am trying to understand why the proof was not stopped at the encircled step. 1. Is there no axiom that says ## x \cdot 0 = 0 ## ? 2. Isn't the sixth...

Dear Everyone, I have some feeling some uncertainty proving one of the axioms for a group. Here is the proof to show this is a group: Let the set T be defined as a set of 2x2 square matrices with coefficients of integral values and all the entries are the same. We want to show that T is an...

In the Zermelo-Fraenkel axioms of axiomatic set theory we find: Axiom. Given any set x, there is a set such that, given any set z, this set z is a member of if and only if every element of z is also an element of x. Why is this needed as an axiom? why isn't it merely a definition? Under...

So apparently the proof involves a trick that converts the problem of a general power set ##\mathscr{P}(M)## of some set ##M## which has of course the property of not having pairwise disjoint set-elements to a problem that involves disjoint set-elements. I do not understand why this trick is...

My discussion (copying this from a Facebook post dated April 29th, 2018): This weakening of the Wightman axioms is not considered in, for example, Section 3.4 of R F Streater, "Outline of axiomatic relativistic quantum field theory", Rep. Prog. Phys. 38 771-846 (1975), where Streater critiques...

So , what I was wondering about was a slight difference in notation, for which I am not certain if correct (mine, in particular.). The induction axiom says: If M is a subset of ℕ, and if holds that: a)1∈M b)(∨n∈ℕ)(n∈M→s(n)∈M) then M=ℕ. Now my question is: why do we write (∨n∈ℕ)(n∈M→s(n)∈M)...

I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?

I was wondering about the following scenario, we have a certain differentiable manifold with the standard topology not induced by any previous metric structure on the manifold. There is no natural way to identify a vector with its dual(no canonical isomorphism between them), If we had to...

It seems that it is self evident. After all, a set A consists of things. These things exist as part of the set. Therefore the idea is that I can think about the item of the set without really thinking about the other items in that set. The set is in a sense the totality of the parts. Why is this...

Following theorems are congruent(a) Axiom of Choice (b) if ∀i:i∈I: <Yi | i∈I > → Yi≠Ø (c) Ø∉S → ∃f: f is on a set S s.t. f(X)∈X for all X∈S. where f is choice function of S. I am confused with the theorem (c), as how the Collection S does not include empty set. I believe every set needs to...

Is it possible to use Axiom of Choice to prove that there would exist a sequence (A_n)_{n=1,2,3,\ldots} with the properties: A_n\subset\mathbb{R} for all n=1,2,3,\ldots, A_1\subset A_2\subset A_3\subset\cdots and \lim_{k\to\infty} \lambda^*(A_k) < \lambda^*\Big(\bigcup_{k=1}^{\infty}...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with Exercise 2.1.29 ... Exercise 2.1.29 (including the Completeness Axiom) reads as...

Homework Statement In Fano's Geometry, we have the following axioms a. There exists at least one line b. Every line has exactly three points on it c. Not all points are on the same line d. For two distinct points, there exists exactly one line on both of them e. Each two lines have at least one...

Obviously information exists. From the concept of information quanta we can create physical models of mathematical concepts. Are ZF axioms redundant?

The hereditarily finite sets(a subclass of the Von Neumann universe) are an axiomatic model that corresponds to the usual axioms of set theory but with the axiom of infinity replaced by its negation(showing its independency from the other axioms of set theory). Some mathematicians (a minority)...

I was reading through the early chapters of Ross' book on analysis in the section covering the completeness axiom. See below. Followed by a few examples. I'm confused as to why in the example (e), the set does not have a minimum. I can understand that it does not have a maximum, but it...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with the...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ... At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

Homework Statement Provide a complete formal proof that ## \vdash ((A \rightarrow B) \rightarrow C) \rightarrow (B \rightarrow C)##. Homework Equations I am only allowed to use modus ponens and these four 'sentential logic' axioms: A1 ## \neg \alpha \rightarrow (\alpha \rightarrow \beta)## A2...

It has been stated that in axiom of regularity , a set cannot be an element of itself and there is a proof for which S={S} . I can understand his proof since S is the only element and hence its method of proof is viable here . But , what if I change the question to S= {S,b} ( it is a set which...

Homework Statement Show, using the axiom of completeness of ##\mathbb{R}##, that every positive real number has a unique n'th root that is a positive real number. Or in symbols: ##n \in \mathbb{N_0}, a \in \mathbb{R^{+}} \Rightarrow \exists! x \in \mathbb{R^{+}}: x^n = a## Homework...

If the axiom of induction was extended to include imaginary numbers, what effect would this have? The axiom of induction currently only applies to integers. If this axiom and/or the well ordering principle was extended to include imaginary numbers, would this cause any currently true statements...

So I understand how to prove most of the axioms of a vector space except for axiom 10, I just do not understand how any set could fail the Scalar Identity axiom; Could anybody clarify how exactly a set could fail this as from what I know that anything times one results in itself 1u = u...

It states that no set can be an element of itself..what does it exactly mean?why can't. We consider the whole set as a single element?

It is said the axiom of QM is observation.. but if observation is secondary effect of more primary dynamics that don't involve observations.. could it still be called QM, and what's it supposed to be called? In other words. Say QM is emergent from a deeper reality.. and we are to study the...

I doubt it but I was doing some work on trying to remove time from Classical Physics (just for the hell of it) and I came across a formula that made me go "huh, not seen that before, but it's kind of neat." Just out of curiosity has anyone seen this formula before? X = √ ((X/2Π) * (X*2Π))

So I've been learning Set Theory by myself through Jech and Hrabeck textbook, and I'm having trouble understanding some axioms. 1. Homework Statement What exactly is the difference between the axiom of pair and axiom of union? From what I understood, the axiom of pair tells us that there is a...

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Foundation which reads as shown...

Homework Statement Here the formulation of Dedekind's axioms that I am using:Suppose that line ℓ is partitioned by the two nonempty sets ##M_0## and ##M_1## (i.e., ##\ell = M_0 \cup M_1##) such that every point between two points of ##M_i## is is also in ##M_i##, for ##i = 0,1##. Then there...

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Infinity which reads as shown...

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Replacement which reads as shown...

Hi guys. So I've been wondering, what's so controversial about the axiom of choice? I heard it allows the Banach-Tarski Paradox to work. A little insight would be much appreciated, thanks.

The speed of light is constant in all frame of reference... So the relative motion of the source of light and the frame of reference from which you are making measurement does not matter... Is it an Axiom in relativity ? I understand that they are experimentally proved concepts... But is it an...

P(x) = x ∉ x ⊃ for any set A, there is a set B such that x ∈ B iff x ∈ A and x ∉ x Does the above mean that different things can bear the same property. For instance, x can be bipedal means x can be an element of the set human or x can be an element of the set ostrich.

Hello, What "choice" does the Axiom of Choice permit us to make? I've searched high and low and not found a satisfactory answer. To me it seems to add no new information to its hypotheses: Given an arbitrary collection of non-empty sets, isn't it true, without assuming anything and just by...

As I understand the ZFC solution to Russell's paradox, since {x|x\notinx} must be {x|x\notinx}\capS for some set S, the paradox goes away, but in Morse-Kelley, if I understand Class Comprehension correctly, although again there must be some M such that {x|x\notinx}\capM, this M may be a proper...

I've already asked somebody through email this question, so I'll copy and paste part of my email: Basically, I'm wondering why doesn't it fall from the other axioms, and if it does in fact not fall from the other axioms (which it apparently doesn't), why the axioms can't be slightly modified...

The Axiom of Dependent Choice, a weaker version of the Axiom of Choice, states that for any nonempty set X and any entire binary relation R on X, there is a sequence (x_n) in X such that x_n R x_n+1 for each n in N. My question is, what would happen if you restricted the relations to...

According to wikipedia, absorption is an axiom for a boolean algebra. This seems incorrect to me, since I believe absorption can be proved from the other axioms (distributivity, associativity, commutativity, complement, identity). Thoughts? ## AB' + A = AB' + A*1 = A(B'+1) = A(1) = A ## BiP

In page 6 of Naive Set Theory by Halmos, he introduces the definition of the axiom of specification, then sets up one example based on the axiom, in which he changes ##S(x)## to ##x \not \in x## to illustrate something. I understand that this mean ##x## doesn't belong in ##x##. Afterwards...

According to a result of Paul Cohen in a mathematical model without the axiom of choice there exists an infinite set of real numbers without a countable subset. The proof that every infinite set has a countable subset (http://www.proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset) is...

The axiom of choice on a finite family of sets. I just been doing some casual reading on the Axiom of CHoice and my understanding of the is that it assert the existence of a choice function when one is not constructable. So if we have a finite family of nonempty sets is it fair to say we can...