All Questions
Tagged with philosophy axioms
56
questions
2
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Confused about abstract models for axiomatic systems
I am studying axiomatic systems and I have a hard time understanding how one is supposed to come up with an "abstract" model for an axiomatic system.
I will use the following example taken ...
- 256
3
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2
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111
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What is the formal system when we are using many different sets of axioms?
I am just starting to learn about formal systems, and have learnt that the many axiom systems in Mathematics, such as those of plane geometry, Peano's axioms, vector axioms, etc. can each be used to ...
- 3,019
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339
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What is the deal with the bizarre philosophy in historical and current axiomatic set theory? [closed]
Many respectable mathematicians have written about "true axioms" or similar concerns about whether all mathematical theorems are in fact "real" or "true". This seems to ...
- 337
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61
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in modal set theory, why it is issue?
I have been studying the Iterative Set Concept within the context of the paper titled "Modal Set Theory" from Menzel specifically on pages 11-12.
"As we’ve just seen, the iterative ...
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What are some good references to help explain the need for an axiom schema of replacement rather than an axiom of replacement?
I’m looking for reading material about the philosophical and mathematical issues that may result from using an axiom schema of replacement rather than an axiom of replacement. Among other things, it ...
2
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1
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504
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Concrete and abstract models of axiomatic systems
In order to prove the consistency of an axiomatic system we must come up with a model. Wikipedia gives the following definition for a model of an axiomatic system:
A model for an axiomatic system is ...
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172
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Why lines and planes as primitive notions?
I'm preparing geometry classes and I thought it is a good time to answer a question I had when I started to study geometry: why, in Euclidean axiomatic geometry, is the notion of a straight line ...
- 837
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116
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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 ...
- 4,310
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152
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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/...
- 140
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85
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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 ...
- 4,631
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250
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Why this formulation is not the official exposition of ZF-?
Specification: if $\phi$ is a formula that doesn't use the symbol $x$, with symbols $``y,a,b"$ as its free variables, then : $$\forall a \forall b: \forall k \exists! x \forall y (y \in x \...
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3
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What do Gödel's incompleteness theorems actually tell us, and how? [duplicate]
I'm currently reading a book on the famous Gödel incompleteness theorems which, at least as I originally understood it, purport to prove that mathematics itself cannot be axiomatized (that is, there ...
- 1,381
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3
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177
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What exactly are axioms?
The title says it all. What exactly are axioms? I mean, I know that is a statement which do not need to be proven. But what are the requirements to be an axiom? For instance, may I state something ...
- 516
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278
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Prove that the incident axioms are independent
Prove that the incident axioms are independent, that is:
Indicate geometry model such that:
b) the l2 axiom does not hold and the l1 and l3 axioms do
I1. For any two distinct points A, B there ...
3
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2
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152
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Should we define something after we have prove that exists based on the axioms?
In last days I have a hard time to understand what a definition is in mathematics. Until today I thought definition had a dual role in mathematics.
Dictionary role The first role is that it mereley ...
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