48
votes
Accepted
Are there more truths than proofs?
That's correct, but it is not very interesting: we don't have "access" to most mathematical facts. It is not even clear what a proof of an arbitrary mathematical fact would mean, because to ...
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16
votes
Accepted
How do we compare the set of natural numbers in different models of ZFC
We cannot compare the $\omega$s in different models, can we? If we say that one is embeddable into another, an embedding $\omega\rightarrow\omega'$ between the two sets of natural numbers in different ...
- 240k
15
votes
Accepted
Nonstandard infinite / hyperfinite sum in IST
Welcome to Math.SE!
You did not indicate how much background you have in Internal Set Theory (IST) - but based on what you wrote, it seem like you have a few misconceptions about it. Let us start by ...
- 9,897
15
votes
Understanding ZFC
ZFC does generate a “theory”. Formally speaking, a theory is simply a set of sentences in the given formal language. Any set of axioms generates a theory, namely the collection of sentences provable ...
- 1,524
14
votes
Are there more truths than proofs?
The argument is fallacious.
We can use the same basic argument in an even simpler setup, which clearly indicates what goes wrong.
There are uncountably many subsets of the natural numbers. There are ...
- 39.8k
14
votes
Is proof of the law of identity a case of circular reasoning?
I don't understand where you think the circularity is coming from. We have an intuition about equality, namely "everything is equal to itself". We'd like to formalize this intuition as a ...
- 74.1k
11
votes
Don't Understand Mechanism behind Proof by Contradiction
For the second case, the argument runs by cases. Roughly, it is:
P1. $A\lor \neg A$, by the principle of the excluded middle.
P2. $A \implies \neg A$, by assumption.
By applying P2 to P1, we have:
L3. ...
- 1,216
11
votes
Accepted
Is FOL really sound?
First let's be clear about the fact that there are two levels at play here.
At the top level (sometimes called the "meta-level"), there is the logical reasoning that we human mathematicians ...
- 74.1k
11
votes
Understanding ZFC
You asked:
If one is to do mathematics rigorously, they would like every concept they mention, such as a function or relation, to have a precise definition. How can we do that with ZFC if ZFC does ...
- 40.2k
10
votes
Accepted
Is there a technical reason to require relation symbols to have positive arity?
Purely historical/conventional. There is no technical reason to exclude $0$-ary relation symbols (proposition symbols).
I can think of two reasons why model theorists tend to ignore proposition ...
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9
votes
Accepted
Is "non-rigid" first-order axiomatisable?
Let $L' = L\cup \{\sigma\}$, where $\sigma$ is a unary function symbol not in $L$. Let $T'$ be $T$ together with axioms asserting that $\sigma$ is a non-trivial-automorphism (non-triviality is $\...
- 74.1k
9
votes
Accepted
Definition of transitivity of relations
The two are not equivalent, and your teacher's (EDIT: per the OP's comment below, they miscopied the definition) proposed definition is wrong. In particular, the property your teacher describes is ...
- 240k
9
votes
Accepted
What exactly is the unique union of a family of sets?
The uniqueness is in the formula used to define the "unique union":
$\{x \mid ∃!A(A∈F \land x∈A) \}$.
This means that if an element $a$ belongs to e.g. two sets $A_1$ and $A_2$ of the ...
- 93.2k
8
votes
Don't Understand Mechanism behind Proof by Contradiction
A long answer
Here I address the second part of your question dealing with the consistency of math. For the first part, I like the answer by Lemmon.
All arguments by contradiction, as well as all ...
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8
votes
Accepted
What is the justfication for splitting up statements and quantifiers?
You could insert the following intermediate steps:
\begin{align*}
\forall x \forall y \left(x \notin X \vee y \notin Y\right)
& \iff \forall x\bigl( \forall y (x \notin X \vee y \notin Y) \bigr) ...
- 117k
8
votes
Is every complete consistent theory satisfiable in ZF?
Note: This answer has been subsumed by my more recent answer. But I've left this older post up (it actually contains two separate answers) since there may be some independent interest in it, and there ...
- 9,387
8
votes
Are there more truths than proofs?
A slightly different perspective - Yes, there are uncountable (trivial) mathematical facts, but you don't need to individually prove each of those facts. What's much more interesting is the fact that ...
8
votes
How do we compare the set of natural numbers in different models of ZFC
Let $(M,\varepsilon)$ and $(M',\varepsilon')$ be models of ZFC. Remember that $M$ and $M'$ are sets (in our ambient set-theoretic universe), and $\varepsilon$ and $\varepsilon'$ are binary relations ...
- 74.1k
8
votes
Nonstandard infinite / hyperfinite sum in IST
You asked several questions that should really be separate posts, but let me start by answering one. You asked:
"If anyone could provide a detailed proof that a sum indexed by an unlimited ...
- 40.2k
7
votes
Accepted
How do I actually use free variables in first order logic?
There is something you're missing, but I might even go so far as to say that this is the fault of the literature.
bullet (1) says that a well formed proposition with free variables does not have a ...
- 800
7
votes
Accepted
Is lambda calculus a sub-system of first-order logic and set theory?
For theories $T_0$ and $T_1$ over the same language, we say $T_0$ is a subtheory of $T_1$ when every theorem of $T_0$ is also provable in $T_1$. If we require theories are closed under deductions, ...
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7
votes
Is 1+1=2 logically equivalent to 99+1=100?
The sentence $1+1=2$ is not logically equivalent with $99 +1 =100$.
To see this note the way logical equivalence is defined in model theory. In first-order logic two sentences $\varphi, \psi$ of some ...
- 1,064
7
votes
Accepted
First-order semantics with minimal metatheory
Essentially, "model" in the usual set-theoretic sense gets replaced by "complete theory with the strong witness property" (for some subtleties around this point, and in particular ...
- 240k
7
votes
Is proof of the law of identity a case of circular reasoning?
There is a difference between $a=a$ and $\forall x[x=x]$. The second one has a quantifier. Of course passing from one to the other is completely trivial, given the introduction rule for the universal ...
- 25k
6
votes
Don't Understand Mechanism behind Proof by Contradiction
Just so you know, what you’re describing is called proof of negation, while proof by contradiction usually means specifically that not-A is assumed, a contradiction derived, and thereby A is proven. ...
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6
votes
Accepted
Reasoning in natural language vs. reasoning in formal language
This question straddles philosophy and mathematics. Or perhaps I should say, it involves both philosophical and technical questions about mathematics. Also it is ripe with history.
On the one hand, ...
- 8,612
6
votes
Accepted
Implicit and explicit definability over $\mathbb{R}$ of the exponential function
I think there are too many details here to write an explicit solution in this answer box. But I'll give you a quick sketch (or, perhaps, an implicit solution?).
$e^x$ is implicitly definable: Use the ...
- 74.1k
6
votes
Accepted
Conjoining models: can we?
This question is (necessarily) somewhat vague, but there's a good sense in which the answer to the question is negative: we can whip up a computable sequence of sentences $\varphi_i$ with a uniformly ...
- 240k
6
votes
Accepted
Confusion about the consistency of $\mathsf{ZFC}+\neg\mathsf{Con}(\mathsf{ZFC})$
Yeah, you're mixing up the levels.
We formalize it as $$ \sf \lnot Con_{ZFC}\to \forall p\in Sentences_{LST}\; Prov_{ZFC}(p),$$ where $\sf Prov_{ZFC}$ is the same provability predicate that goes into $...
- 56.8k
5
votes
Accepted
(Fake proof) How are axiom schemas not "set theory in sheep's clothing"?
Any FOL theory with computably enumerable axiom schemas is essentially equivalent (i.e. is bi-interpretable) with another FOL theory with finitely many axioms. You simply encode the (finitely many) ...
- 56.6k
Only top scored, non community-wiki answers of a minimum length are eligible