0
votes
2answers
71 views

Does axiom of foundation/regularity protect against Russell-like paradoxes?

In ZF set theory the axiom of regularity (also called axiom of foundation) says that: In all nonempty sets x there is an element y such that x∩y=∅ As I been told that the intention of the axiom ...
-1
votes
1answer
85 views

Does Mathematics exists apart from the mathematician? [on hold]

Does Mathematics exists apart from the mathematician? Explain yourself. Mathematics seems to be a projection of the mind. But from where the mind originates? Can the source of the mind be known or you ...
-1
votes
0answers
247 views

Are paradoxes a threat against mathematics? [closed]

Or are they just mathematical tools? Given a binary predicate $R$, written in infix notation, there are unitary predicates $p$ such that: $\qquad \exists y \forall x: x R y \leftrightarrow p(x)$. ...
2
votes
1answer
74 views

Building math theory on absurd axioms - reducing math to logic

I know similar questions have been asked and i know my terminology might be wrong but I am trying to come to an answer to whether math can be derived from logic. Wikipedia defines logic as use and ...
3
votes
5answers
148 views

Is $'' \sum_{n = 1}^{\infty} (-1)^n \; \text{is a real number}''$ an invalid statement or a false proposition?

So we're beginning an introductory logic course and my professor is giving examples for valid statements/ propositions - meaningful statements that are either true or false but not both. So he puts ...
2
votes
2answers
127 views

Three-valued logic as foundation

Isn't it more natural to use Three-valued logic(false-true-unknown) as the foundation of mathematics? It is a better model for natural languages. And it also can model sentences like the lair paradox ...
3
votes
2answers
143 views

Gödel's incompleteness theorems

In the last paragraph of Stephan Hawking's speech "Godel and the End of the Universe", he mentioned "... I'm now glad that our search for understanding will never come to an end, and that we will ...
2
votes
7answers
450 views

Are the real numbers really uncountable?

Consider the following statement Every real number must have a definition in order to be discussed. What this statement doesn't specify is how that loose-specific that definition is. Some examples ...
0
votes
2answers
119 views

Is the “Most Important Property a Set S has” Necessary and Sufficient to Define a Paradox-Free Notion of Set?

About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a ...
2
votes
1answer
105 views

Why didn't Frege succeed in his attempts to reduce mathematics to logic?

My background: Sophomore-level understanding of mathematics and philosophical logic. All the explanations I have found online so far are either far too technical or too simplistic. Thanks in advance ...
0
votes
1answer
56 views

particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
4
votes
4answers
383 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
0
votes
0answers
42 views

The Major Weaknesses in Ramified Type Theory

I am reviewing a paper on the major weaknesses of Ramified Type Theory in predicative second-order arithmetic. These four are listed as "weaknesses." But, I have my doubts. It seems at least that 3) ...
2
votes
2answers
157 views

Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, ...
6
votes
2answers
251 views

Set theoretic realism

What are the main contemporary arguments for and against realism about set theory?
1
vote
2answers
100 views

Abstract Objects in Logic

I am confused on the concept of extensionality versus intensionality. When we say 2<3 is True, we say that 2<3 can be demonstrated by a mathematical proof. So, according to mathematical logic, ...
1
vote
3answers
105 views

Why do we formalize conceptions?

Why do we always try to formalize conceptions? Let's take the naive conception of sets, why do we try to write down a list of axioms? what do we earn in doing so? I'm looking especially for ...
2
votes
2answers
85 views

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works.

Study of all published works of Bertrand Russell on foundations of mathematics: Please recommend his works. I think Bertrand Russell was a special mind and I set a goal for myself to study all his ...
1
vote
0answers
43 views

Trying to understand Hintikka's logic of Knowledge and belief

I try to understand Hintikka's logic of knowledge and belief but am a bit stumped by it. I study " Knowledge and belief , an introduction to the Logic of the two Notions", (Kings College ...
1
vote
1answer
48 views

Constructivist Interpretation of a Function

Lets suppose I have an exponential function $a^{x}$, and I desire to show that for any number $n$ in $(0, \infty)$, it is possible to find a value of $x_0$ such that $a^{x_0} = n$. The simplest proof ...
7
votes
1answer
210 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
4
votes
3answers
170 views

Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?

It is frequently stated (in textbooks, on Wikipedia) that the "Law of large numbers" in mathematical probability theory is a statement about relative frequencies of occurrence of an event in a finite ...
3
votes
6answers
195 views

Logical issues with the weak law of large numbers and its interpretation

In several probability textbooks I have found what amounts to the following argument: Let A be an event in some probabilistic experiment. Let p=P(A) be the probability of this event occurring in ...
1
vote
2answers
77 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
3
votes
2answers
86 views

How does undecidability of 'theoremhood' imply that human ingenuity is necessary in mathematics?

In Robert Stoll's "Set Theory and Logic", there is the following passage on effectiveness of theorems (p. 375) : Mathematical logicians have shown that for many interesting axiomatic theories ...
1
vote
1answer
59 views

What exactly does $\vdash_T G_T \leftrightarrow \lnot \exists y$ Prf$(\ulcorner G_T \urcorner, y)$ mean?

To me this translates to: $G_T$ is provable in $T$ if and only if there doesn't exist a $y$ such that $y$ is a witness to the provability of $\ulcorner G_T \urcorner$. But I'm not entirely sure what ...
3
votes
1answer
93 views

Theorems that we can prove only by contradiction

While most of the world is fine with proofs performed by contradicting the thesis, direct proofs are sometimes considered more elegant than indirect ones. Those who prefer intuitionism or ...
3
votes
3answers
150 views

Implications and Ordinary language

I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources. One particular case is sentences in the form ...
5
votes
2answers
332 views

Who stole the axioms in Natural Deduction?

The study of Gentzen's sequent calculus give me the opportunity to propose some reflections about the concept of logical truth. I'll refer to the english edition of Gentzen's works : The collected ...
0
votes
2answers
97 views

About the concept of logical truth

From Frege and Russell to modern mathematical logic textbooks, there were a "shift" of focus from the concept of logical truth, through that of valid formula, to the current concepts of logical ...
5
votes
0answers
134 views

Gödel's Completeness Theorem and logical consequence

At the end of a long process of "rumination" on "old" math log textbooks, I've found the "missing link" - from my personal point of view - between some issues I've raised in the previous months : (i) ...
0
votes
1answer
25 views

Reformulation of Theories

Philosophical questions (or even just a matter of taste) regarding some mathematical constructions can give rise to reformulations of whole theories, for example, we can develop (Non-standard) ...
12
votes
2answers
201 views

Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
2
votes
2answers
165 views

Again about McGee objections to modus ponens

I would like to "reopen" the previous post regarding Modus ponens because, frankly speaking, I'm not satisfied with some (most of ?) answers by the mathematicians community. Disclaim: I'm not aiming ...
1
vote
3answers
153 views

When and where the concept of valid logic formula was defined?

I was stimulated by a recent question about Gödel Completeness Theorem. All my citations are from Jean van Heijenoort (editor) From Frege to Gödel A Source Book in Mathematical Logic (1967). Gödel's ...
2
votes
1answer
108 views

Why can't ✳1.1 be expressed symbollically in Whitehead and Russell's PM?

✳1.1. Anything implied by a true elementary proposition is true. Pp. In the follow passage, it says, "we cannot express the principle symbolically, partly because any symbolism in which p is ...
3
votes
2answers
85 views

Is Paraconsistent Negation Really Negation?

Let a logic be paraconsistent, if $\phi \wedge \neg \phi \not \models \psi$ for some $\phi, \psi$ (where $\models$ is the logic's consequence relation). There are different ways to prevent a ...
14
votes
3answers
314 views

Mathematical Notation and its importance

You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...
5
votes
4answers
398 views

What is the “correct” reading of $\bot$?

I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus. In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14] ...
7
votes
3answers
581 views

Are all mathematical statements true or false?

I would like to know whether it can be possible for a statement to be neither true nor false. Consider the age old paradox. "This statement is not true" Clearly it cannot be true. If it is false. ...
4
votes
4answers
230 views

What is the difference between asserting “$\phi(a)$” and asserting “$\phi(a)$ is true” in Whitehead and Russell's PM?

The first edition of Principia Mathematica clearly distinguishes "Socrates is a man" and "'Socrates is a man' is true." Judging from the context, the distinction is neither a primitive idea nor a ...
4
votes
1answer
86 views

What is the intuition behind $\Delta_1^0$ sets and $\Delta_1^1$ sets?

In the context of first-order arithmetic, if $\phi$ is a formula with only bounded quantifiers, then if you put existential quantifiers in front it becomes a $\Sigma_1^0$ formula according to the ...
2
votes
1answer
161 views

What's with conditionals in mathematical logic?

Having a bit of difficulty understanding the conditional ($\rightarrow$) in mathematical logic. I read up on the already-existing questions and it did help me understand it better (the 'promise' ...
6
votes
1answer
127 views

Do second-order categoricity proofs require a background concept of set?

In his article "The Set-Theoretic Multiverse", Joel David Hamkins (as part of his reply to Donald Martin's argument that the set-theoretic universe is unique, found in "Multiple Universes of Sets and ...
1
vote
0answers
64 views

Are there intensional classes independent of the set universe?

The hereditarily finite sets can be regarded as purely extensional sets. Furthermore, they are quite independent of the underlying set universe (at least if we look at them from an extensional point ...
5
votes
1answer
78 views

What is $M_x$ in Frege's Basic Law IIb?

Gottlob Frege's magnum opus, "The Basic Laws of Arithmetic" (Die Grundgesetze der Arithmetic in German) constitutes one of most impressive and meticulous attempts at developing a rigorous foundation ...
1
vote
1answer
251 views

Abductive v.s. inductive reasoning

To me, abductive reasoning and inductive reasoning are very very similar, in that they both go from the specific to the general and they are distinguished only through the examples which are provided ...
3
votes
1answer
111 views

Can equinumerosity by defined in monadic second-order logic?

Two properties (or concepts) $F$ and $G$ are said to be equinumerous if they have the same cardinality, i.e. if they can be put in one-to-one correspondence with each other. This can be very easily ...
12
votes
1answer
366 views

What underlies formal logic (or math, generally)?

I read a useful definition of the word understanding. I can't recall it verbatim, but the notion was that understanding is 'data compression': understanding happens when we learn one thing that ...
3
votes
2answers
247 views

Leibniz' Law and that good old riddle

There exists a Theory of Identity in mathematical logic. I've encountered it for the first time in Principia Mathematica by Alfred North Whitehead and Bertrand Russell (1910). Quote: "This definition ...