Questions dealing with set-theoretic functions defined by transfinite recursion.
Questions dealing with set-theoretic functions defined by transfinite recursion.
Questions tagged [transfinite-recursion]
166 questions
170
votes
11 answers
Do we know if there exist true mathematical statements that can not be proven?
Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.
Phrased another…
Jeremy
- 1,591
21
votes
3 answers
How to apply the recursion theorem in practice?
The recursion theorem
In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set $X$, $a \in X$ and a function $f \colon X \to X$, the theorem states that there is a unique function $F:\mathbb{N} \to X$ …
Dan Christensen
- 15,710
12
votes
3 answers
What does $\upharpoonright$ in $G(F\upharpoonright\alpha)$ mean?
More formally, we can state the Transfinite Recursion Theorem as follows. Given a class function $G\colon V\to V$, there exists a unique transfinite sequence $F\colon\mathrm{Ord}\to V$ (where $\mathrm{Ord}$ is the class of all ordinals) such that…
11
votes
2 answers
"Transfinite Peano Axioms"
Perhaps, the class of ordinals $\Omega$ can be axiomatised up to isomorphism by claiming it to be well-ordered such that for every subset $X\subseteq \Omega$ there exists a "succesor" ordinal $\sigma$ which is the smallest ordinal larger than any…
Lucina
- 697
9
votes
2 answers
Is transfinite induction needed to remove all the elements from an uncountable set?
Given any uncountable set S, would I need to use transfinite induction to prove if I remove single elements recursively, I will be left with the empty set?
It seems like this can be thought of as an arbitrary intersection problem, so just a logical…
Just Some Old Man
- 1,341
8
votes
1 answer
How to prove an extension of ZFC is conservative
Working in ZFC.
I've defined a function-like binary predicate $R$ on a proper class. It has to be recursive; i.e. $R(a,b)$ must usually depend on one or more $R(c,d)$ for some $c$s and $d$s calculated from $a$ and $b$. There is, however, no…
Neil Toronto
- 577
8
votes
1 answer
Contents of Gentzen's consistency proof of PA
It is surprising that there is lack of information on Gentzen's consistency proof - sure, there are some contents on Gentzen's first consistency proof of Peano axioms, but not on what we usually say Gentzen's consistency proof. Thus the question:…
Brimos
- 179
7
votes
1 answer
Direct proof of principle of transfinite induction
This is a problem from the book Set theory by You-Feng Lin.
Principle of Transfinite Induction
Let $(A,\le)$ be a well-ordered set. For each $x \in A$, let $p(x)$ be a statement about $x$. If for each $x \in A$, the hypothesis "$p(y)$ is true for…
nomadicmathematician
- 6,309
7
votes
1 answer
Definition by Recursion
I just started studying logic, not as a course at a university, but as pastime. Since I do not study logic at an institution I use many different textbooks, including Enderton's $A$ $Mathematical$ $Introduction$ $to$ $Logic$ and Van Dalen's $Logic$…
user92805
- 71
7
votes
1 answer
TREE(3) and the Goodstein sequence
TREE(3) is an extremely large number that requires ordinal arithmetic to prove it is finite. For what value of n would $G(n)>TREE(3)$? The length of the
Goodstein sequence $G(n)$ is how many numbers in the sequence beginning with n are traversed…
Sheldon L
- 4,599
- 1
- 18
- 33
7
votes
1 answer
Is there a generalization of transfinite recursion that allows defining proper classes?
Transfinite recursion lets one define a sequence of sets $S_\alpha$, for $\alpha$ an ordinal. My question is whether it is possible to generalize this recursion to allow $S_\alpha$ to be classes, not just sets. I would be thankful for a reference to…
Tom
- 1,198
7
votes
2 answers
Application of Transfinite Induction
Our teacher gave us for practice to prove some properties of $V(\alpha)$ defined as
$$V(0) = \emptyset,\; V(S(\alpha)) = \mathcal{P}(V(\alpha)),\; Lim(\alpha): V(\alpha) = \bigcup\{V(\beta)\; |\; \beta < \alpha\}$$
The properties are
1) $\alpha <…
7
votes
1 answer
Epsilon induction
I recently came upon this technique called epsilon induction, and was searching for some proof using the same. But I saw no such proof. Does someone know of any proof using this technique?
Ylva
- 71
- 1
6
votes
1 answer
Classifying Vector Spaces without AC
Using the axiom of choice we can give a simple classification of all vector spaces over a given field $K$ up to isomorphism: Any $K$-vector-space $V$ is just isomorphic to $\bigoplus_{i\in B}K$ where $B$ is a basis for $V$. Given AC we even know…
H.D. Kirchmann
- 183
6
votes
2 answers
Showing there is only one isomorphism between well ordered sets using transfinite induction
I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do this I want to show by induction that there is only one…
Serpahimz
- 3,971