# Questions tagged [formal-systems]

A formal system is broadly defined as any well-defined system of abstract thought based on the model of mathematics. (Def: http://en.m.wikipedia.org/wiki/Formal_system)

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### Why Löwenheim–Skolem theorem asserts the non-existence of such predicates in 1st order logic

Suppose there was a predicate, in the language of 1st order $\mathsf {PA}$, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it ...
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### Meaning of Provable recursiveness

Is there any difference between provably total function and provable recursiveness of a function in first order PA ? From provably total I mean that the totality of the function itself is provable in ...
1 vote
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### How to prove the property of being a tautology is hereditary under the Rule of Substitution?

In Appendix of Godel' proof by Nagel, there is a proof being left to the reader. The proof is how to prove the property of being a tautology is hereditary under the Rule of Substitution? I have no ...
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### Circularity in the argument that Gödel's incompleteness theorems undermine Hilbert's program

I'm only familiar with the very basics of mathematical logic, but over the last few days I have been looking into Gödel's incompleteness theorems and it seems to me (but I might simply be grossly ...
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1 vote
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### Can Bayesian analysis address challenges posed by Gödel's incompleteness theorems?

In short, according to Gödel's incompleteness theorems, if a formal mathematical system is consistent, then it will contain true statements that can neither be proven nor disproven it will be unable ...
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### Connection of axioms of first order logic and axioms of first order theory

If we have a set of sentences S in first order logic. We know that we can create a first order theory Th(S) from S, which is the "set S" union "the sentences which we can prove them ...
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### Axiomatic system that differentiates between straight lines and curved lines

I was thinking about a formal system that involves a set of primitives along with a property that, when applied to the model of planar one-dimensional objects, can distinguish between straight lines ...
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### ((())()) is a theorem of PR?

If we have the following formal system, called PR whose formulas are strings of well-formed parentheses. The language has two symbols '(' and ')'. Any expression is a formula. The only axiom is (). ...
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### Gödel on the “True Reason” for Incompleteness

In footnote 48a of his famous paper on incompleteness, Gödel writes: [T]he true reason for the incompleteness inherent in all formal systems of mathematics is that the formation of ever higher types ...
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### Do Tarski's axioms apply to higher dimensions?

I came across the wonderful fact that the theory of Euclidean geometry in 2 dimensions is complete, consistent and decidable — as shown by Tarski's axiomatization. I know very little about this. My ...
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### Can ZFC be proven from weak systems using consistency of those systems?

Tl;dr Can we take a weak system $A_0$ then show $$A_0 + Con(A_0)\implies Con(A_1)), \space A_1 + Con(A_1)\implies Con(A_2)), \space A_2 + Con(A_2) \implies \dots$$ terminating in ZFC? My understanding ...
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### Representing first order sentences as conceptual graphs

Here are three first order axioms that represents a part of a mereology theory. Reflexivity $\forall x : part(x,x)$ Antisymmetry $\forall x \forall y : ((part(x,y) \land part(y,x)) \implies (x = y))$ ...
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### Is the decidability of all possible axiomatizations equivalent to decidability?

Introduction This is a weird take on a few different topics in logic, so bear with me. To frame this question, consider that "axiomatization" is to "theory" as "...
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### How do we classify the obvious fact that $A \xrightarrow{gf} C$ can be decomposed into two arrows? Is it just notational?

The following image is of an app I'm working on called "Abstract Spacecraft". It will allow you to make logical rules and also apply them to diagrams at the click of a button, should the ...
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### Generalization of nonexpressibility of solutions

Are there any ideas/areas of math that generalize the idea of when a solution to a question can be expressed in some given formal system? Something that (at least in spirit) covers results like No ...
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### Help with metalogic induction proof involving string parsing

I have a formal language (basically language of first order logic) in which Wffs are defined by following. String $A$ is a Wff if it satisfies one of: $A$ is atomic $A \equiv (\lnot B)$ and $B$ is a ...
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### Do we have to resort to intuition at the very root of our thinking?

Formal first order logic is the foundation of ZFC set theory. This gives me the impression that a theoretical system has to be based on a formal logical system, with its own axioms and deduction rules....
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### Why isn't there an app that allows you to enter in all the rules of a given formal system so that the app supports all formal systems of math?

I'm jumping around between articles about ETCS to Simple TT to Calculus of Constructions wondering what my app should focus on. I'm wondering, why there isn't yet a software app in which you can ...
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### Can you give an example of a non-formal system?

A formal system is defined as an abstract structure used for inferring theorems from axioms according to a set of rules (from Wikipedia). Language, logic and mathematics are considered as formal ...
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### Essential undecidability of binary string arithmetic

The weak theory of concatenation of binary strings is essentially undecidable.1 Is Presburger arithmetic with two successors (one for each letter) essentially undecidable? Formally, consider the ...
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### Precise statement of the syntactic category of a logic/type theory, in maximum generality?

I've been trying to understand the notion of syntactic category for a type theory/logic. This entry in the ncatlab is the closest I've found to a clear explanation. It seems like a fairly good article,...
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### Simplest set of axioms and inference rules for first-order logic?

I'm interested in finding the "simplest" formulation of first-order logic. To be precise, what formulation of first-order logic has the fewest total axioms (or axiom schemas, henceforth just ...
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### How to write "the set of all walks of a graph" in formal logic notation?

I have this question on the CS stackexchange to figure out how to model "a walk of a graph" in a custom formal language. I think the problem is that there is not just one walk of a graph but ...
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### How do you formalize this reasoning?

There is this simple number theory problem which says "how many 4 digit numbers that are divisible by 3 and whose digits exclude 2, 4, 6 and 9 exist?" the solution is quite intuitive: each ...
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### Axiom scheme for mathematical induction in formal axiom system for Peano Arithmetic

I'm reading the book "Gödel's incompleteness theorems" by Smullyan. (I found it online here: https://isidore.co/calibre/get/pdf/5823). In Chapter III he explains the Axiom System for Peano ...
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### What is a precise definition of soundness?

I'm trying to better understand soundness, especially in contrast to semantic consistency. Here is what I've put together so far: Soundness: A theory is sound if all theorems are true under all ...
1 vote
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### Is it possible to create a "hack-proof" system?

Various organizations, such as DARPA, have purported to create virtually "hack-proof" software systems using mathematical techniques such as formal verification. In short, the idea is ...
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### Is there any technical, mathematical or logical, terminology for a set of propositions, all of which cannot be True at the same time.

As the question states, what is the technical term (if any) for a set of propositions which cannot all be true at the same time. That is to say, If there is a set, $A${x1, x2, x3, ... , xn} for which ...
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### Is it possible to list all hidden lemmas of a proof?

I'm studying Imre Lakatos' Proofs and Refutations for my master's thesis. Currently I address the concept of hidden lemmas, which I understand to mean unstated assumptions of a mathematical proof, ...
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### Beta Reduction Constraints

The definition of $β$ reduction is the following : $$(λx.M)N \rightarrow_{β} Μ[x∶=N]$$ So basically we stop treating $x$ as a bound variable and we perform substitution of the now free variable $x$ ...
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I came across the definition of beta reduction in Lambda Calculus which is : $$(λx.M)N \rightarrow_β Μ[\space x:= N \space]$$ under the constraint that the $FV(N)$ are still free after the ...