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166 views

How do models determine truth values if the external theory is incomplete?

I'm currently learning model theory from Chang and Kiesler's Model Theory: 3rd edition. Something about the basic relationship between syntax and semantics is troubling me. The book describes ...
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Numbers/cardinal numbers [closed]

so i had two classes this semester abstract algebra and philosophy of math. I enjoyed them both but there is one thing that i have been trying to understand all the semester and just can't get ...
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830 views

Can we make intuitionistic logic "intuitive"?

This is something I am wondering about, because all the formulations I've seen of the logic seem fairly difficult to grasp, e.g. lists of abstract axioms that have a few missing versus classical logic....
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1answer
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Ultra-finite math [closed]

Assume our universe is finite in any sense (the information amount was some $10^{120}$ bits, if I remember correct) and deal with the following, slightly (and intendedly) preposterous, scenario: The ...
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83 views

What doe the term 'object' in the definition of a set mean?

What exactly the word 'objects' does mean in the definition of a set, is it an indefinable concept? then how can we say it is a 'well defined object' ? Can I say 'objects' are 'being that exists' in ...
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Histories in Scott-Potter set theory

I have been reading Michael Potter's Set Theory and its Philosophy. I am confused by the concept of a history, which I understand is somewhat unconventional. The definition given is as follows: The ...
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57 views

Does the existence of an indirect proof imply the existence of a direct proof?

So direct proofs are often considered more informative than indirect ones. This got me thinking, and as far as I believe this question has no practical implication - does the existence of an indirect ...
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1answer
40 views

Why can geometric figures, such as a straight line move?

A Cartesian plane is just a set of points. Among those points are some that constitute a straight line. Let $L$ denote such a set of points that constitute a line and $A$, $B$ denote two distinct ...
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Reference of language aspects of Mathematics.

I occasionally wonder about the language aspects of Mathematics. Like when I read Vilém Flusser, Tractatus Logico-Philosophicus by Wittgenstein and Godel's Proof by Ernest Nagel and James Newman. Some ...
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Why some say virtually all of mathematics can be done with FOL when even the class of topologies is not axiomatizable in FOL?

The question can be seen as: please explain the apparent contradiction (the paradox) between the answers in Can we express the theory of a single topology as a multi-sorted theory? and in (Why) is ...
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The correct physical interpretation of Binomial distribution and bernoulli trial in this example

We know that every random variable can have a probability distribution. Examples include the number of heads in many tosses, or the number of ones on a dice after many rolls and so on. Suppose we use ...
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What is the fundamental difference between choosing a ball and rolling a die type of problems in probability?

Suppose, I have a box where I have $n$ balls out of which $b$ are blue. Hence, the probability of picking up a blue ball at random is $p=\frac{b}{n}$. Now suppose, I know the total number of balls, ...
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Heyting's Definition of Spreads and Species

I was reading "Intuitionism - an Introduction, A. Heyting" but I could not understand his definitions of "species" and "spreads" (and also there is a thing which he calls ...
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1answer
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Is it possible to create my own system of math with a new set of axioms that may or may not be observable in "the real world"?

If the axioms that we know about are true statements that can not be proven and are the foundation of "standard mathematics", would it still be considered mathematics if I create my own set ...
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Philosophical/Historical question on the word "kernel" in Algebra

My question is little philosophical/historical, and it came to mind due to some natural curiocity. The word kernel appears in many contexts in abstract algebra. (It also appears in other branches of ...
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Real world relevance of uncountability of R [duplicate]

This may seem like a naive/stupid philosophical question so I am prepared to get destroyed here but what is the real world relevance of the uncountability of the real number system? I understand that ...
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1answer
104 views

Are there infinitely many proofs of every true mathematical statement? [closed]

If we assume there is no limit to the amount of mathematics we create/discover, is it possible that there are infinitely many proofs about any true mathematical statement? For example, we have Wiles' ...
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60 views

Grounds of Stellar Resolution (Transcendental Syntax)

Transcendental Syntax is a program proposed and started by Girard aiming for logic (linear logic, in particular) reconstruction. The idea is to go the "opposite way", forget all the logic ...
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2answers
81 views

Are provable statements true?

I recently started reading an introductory logic textbook, and I haven't got yet to the chapter that talks about completeness theorem, but I just couldn't wait to read about it using shortcuts. I just ...
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links between real and complex analysis

I am new to analysis and just starting to appreciate it. So far I have encountered 3 what seem like fundamental links between the cases of $\mathbb{R}$ and $\mathbb{C}$: The Cauchy-Riemann equations ...
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1answer
161 views

Understanding the natural numbers and Peano's axioms

I am reading the book Analysis $I$ by Terence Tao, where the following is written: [...] $\mathbf{N}$ should consist of $0$ and everything which can be obtained from incrementing: $\mathbf{N}$ should ...
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1answer
74 views

Is it possible for a set to contain non-mathematical objects?

This might be a somewhat philosophical question and perhaps I even have a wrong understanding of what I write as a premise, so I am sorry if that is the case. A set is usually any collection of ...
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1answer
68 views

Would it be possible to exist different mathematical notations for numbers with different properties?

I was reading about how the hindu arabic notation (the notation we use worldwide for numbers) of numbers was benefic for mathematics. Because it makes much easier some operations, such as ...
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Self-referential foundations

A standard way to approach formalization (judging from my experience) is to set up a kind of informal metatheory with few foundational concepts, such as the metalogical implication $\Rightarrow$, ...
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317 views

Is there a mainstream school of thought in mathematics? [closed]

Some years ago(before I started studying mathematics) I was under the impression that most mathematician "believe" in Mathematical Platonism, however, the more I study the more I feel this ...
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1answer
88 views

Is there an equivalent to the Stanford Encyclopedia of Philosophy for mathematics? [closed]

If not: where do you go when you want a short, accessible overview of an area of research?
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Iterative conception: Boolos and Von Neumann universe

I am reading Boolos' famous article "The iterative conception of set" from Benacerraff and Putnam's collection of papers "Philosophy of Mathematics". At page 493 (2nd edition) ...
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630 views

In classical predicate logics, why is it usually assumed that at least one object exists?

In classical predicate logic it is commonly assumed that the domain of objects is non-empty. This validates inferences such as $$\forall x Fx \models \exists x Fx$$ as well as, if the identity ...
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Reference books for Foundation of Mathematics (Logic and Philosophy)

I am an undergraduate student. And I want to build solid foundation for Mathematics. I tried google search but could not get proper recommendation. Please suggest books which covers the subject nicely....
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What formal logic has the smallest metatheory?

I'm currently studying A Concise Introduction to Mathematical Logic (Third Edition) by Wolfgang Rautenberg. What sparked my interest in logic is my interest in foundations in general, so I was ...
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1answer
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Is there a mathematical definition of mathematics?

First of all, I apologize if this question is inappropriate for math SE. In mathematics textbooks, there are defined all sorts of things, like groups, fields, boolean algebras, turing machines, etc. ...
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248 views

Philosophy of forcing

There are usually two expository styles on forcing: internal (forcing over the universe $V$) and external (forcing over a ctm $M$); I guess whether to use general poset $P$ or boolean completion $B$ ...
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3answers
115 views

Possible interpretation of real numbers as functions? [duplicate]

At the end of the day, a real number can be viewed simply as a function over the integers —> the naturals which tells you the digit as that ten’s place (assuming base ten)? You could augment this ...
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1answer
116 views

Are numerical (and/or discrete) solutions to problem equivalent to analytic solutions? [closed]

Are numerical (and/or discrete) solutions to problem equivalent to analytic solutions? If one assumes that all problems must be at least numerically solvable, but not necessarily analytically? Are ...
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1answer
77 views

how universal is Conway's game of life? is it reasonable to expect that a technological alien civilization would recognize, say, a glider?

This is a philosophical one, so apologies if it's not appropriate. I can think of several reasons that Conway's Game of Life would be rediscovered by any mathematically inclined biological life forms. ...
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Is time ever represented as something else than a 1 dimensional line in mathematics?

Is time ever represented as something else than a 1 dimensional line in mathematics? If so, what are the purposes or applications of representing or modelling time as something else than an 1 ...
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1answer
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Defining a set connotatively or denotatively.

“It is said that a set can be defined connotatively or denotatively. Which of these terms applies to the definition by roster and which to the definition by a defining sentence?”- p. 137, Elements of ...
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Why is it that most mathematical statements that mathematicians tend to study are decidable?

This is a bit of a philosophical question. Due to Godel, we know that there are undecidable statements in ZFC set theory. But why is it that most statements that mathematicians tend to study in ...
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Gödel's proof: What if all axioms of a formal system are Gödel sentences

By proof, we know that Gödel's first Theorem applies to certain formal/axiomatic system, while the unprovable statement to which Gödel refers, the so-called "Gödel Sentence", is designed to ...
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Correct general conception of sum and product?

Can, in a very general sense, a sum be viewed as combination without interaction, and a product as a combination with interaction/interdependence? In all "kinds" of addition, the result is ...
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1answer
132 views

Is there a category (or rather a mathematical theory) for which we know a lot about, but not whether its object class is empty or not?

this is a bit of a vague question so let me describe a bit what motivates it: Yesterday I was reading the Wikipedia article about perfect numbers, where I find the section https://en.wikipedia.org/...
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How to prove the consistency of a collection of axioms?

Is there a way to prove the consistency of some chosen axioms? In the two senses following: In each mathematical logic book, there is a special kind of deduction system, which include some logical ...
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1answer
111 views

Name of these lemmas in set theory

Lemma 1.2 If $S$ is countable and $S'\subset S$, then $S'$ is also countable Lemma 1.3 If $S'\subset S$ and $S'$ is uncountable, then so is $S$. I was wondering if there was a name for the logic/...
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How can we be sure that axioms are sufficient?

My question by itself is hard to ask and hard to answer. I don't mean to be ambiguous, but it is one of those philosophical questions that requires a bit of specification. Let me provide some context. ...
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multiplicative number base?

Imagine a number base where instead of writing the number of products for each power of your base (e.g. in decimal, $152 = 1 \times 10^2 + 5 \times 10^1 + 2 \times 10^0$), you had one where you write ...
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Learning Mathematics: unique knowledge of math.

I've come to a point where I now more fully (like 99%) understand that math is about specialized/unique knowledge which can't be summarized by some grand scheme or united principle. So my question is, ...
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900 views

Why don't mathematicians introduce intuition behind concepts as physicists do?

First of all please don't be angry - if anyone might be - and thoughtlessly downvote this post. I'll make it clear that I'm not here to criticise mathematicians - but rather to understand. I ...
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Is "proof by counterexample" a legitimate proof?

Suppose that I am proving the following proposition: Proposition 1. For some $a$, $f(x)=ax^2$ is not an increasing function. I "proved" the proposition by providing a numerical ...
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Does the following principles capture the modern standard line of set theory?

I think that the following axioms describes what modern set theory is all about (on top of mono-sorted first order logic with equality and membership) Extensionality: Two sets with the same elements ...
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161 views

Morphism composition is the property of morphism or the structure of category?

In the definition of category, there is a morphism composition law. If A, B, C are objects, and if f is a morphism from A to B, g is a morphism from B to C, then there is a corresponding morphism from ...

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