2
votes
2answers
53 views

How come that two inductive subsets can be different

In Enderton's "Mathematical Introduction To Logic". Author says that if we have two operations $f(x,y)$ and $g(x)$ and two sets $B$ and $U$ such that $B \subseteq U$. We say that $S \subseteq U$ is ...
1
vote
0answers
58 views

Generalization of simple and transfinite induction

Definition For a set of ordinals $\boldsymbol\alpha$ and ordinals $\gamma$, $\beta$, let $$\boldsymbol\alpha \xrightarrow[\gamma]{}\beta$$ symbolize the proposition that ...
4
votes
1answer
49 views

Inductive definition with choice for sequence

In topology there is a very common way to define a sequence. This usually go something like: "Define $\{z_{n}\}$ to be a sequence such that $z_{0}$ is <blah blah blah>, and $z_{n}$ is such that ...
2
votes
2answers
130 views

Formulation of the (axiom of) mathematical induction

Consider the two following formulations of the principle of mathematical induction. First formulation Let there be given a set $S$ with the following properties : (i) 1 $\in S$, (ii) $n \in S ...
0
votes
1answer
63 views

How do i prove there exists a function $F(n+1)=f_n(F(1),…,F(n))$?

Let $X$ be a set and $f_n:X^n \rightarrow X$ be a function for all $n\in \mathbb{N}$ and $c\in X$. How do i prove that there exists $F:\mathbb{N} \rightarrow X$ such that $F(n+1)=f_n(F(1),...,F(n))$ ...
1
vote
1answer
107 views

Why do sequences exist? What does “constructing a sequence” mean formally?

Everybody knows arguments like: "We can construct such a sequence inductively. Let $a_0$ be chosen as [..]. Then we can choose $a_{k+1}$ out of the set $A_{k+1}$ (which was shown to be non-empty)." ...
1
vote
0answers
96 views

What is “Transitive induction”?

In the book "Artinian Modules Over Group Rings" By Leonid Kurdachenko, Javier Otal, Igor Ya Subbotin (also see http://books.google.com/) one can read (on p.117) "applying transitive induction, we ...
2
votes
2answers
147 views

Finite Set Induction

Let A be a set and let $FS(A)$ be the set of all finite subsets of $A$. Then to prove a formula of the form $$(\forall S \in FS(A))(Q(S))$$ it is sufficiently to prove the following two formulae: ...
7
votes
2answers
161 views

Does this require transfinite induction?

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 ...
2
votes
2answers
234 views

ZF+Induction is Inconsistent?

I am interested in mathematical systems where first order induction (IND) fails. One example is ring theory + $\forall x(Sx=x+1)$. Non-commutative rings are models of this theory. Induction proves ...
5
votes
1answer
113 views

Induction over all ordinal numbers.

I'm starting to learn set theory. I've already learned transfinite induction and I started to wonder if we could do the same on a well-ordered proper class, like the class of all ordinal numbers. It ...
3
votes
1answer
110 views

Can An Axiom Schema be Independent?

Consider the following theory: Ring Theory (RT) + $\forall x(Sx=x+1)$ + first order induction (Ind). The finite rings $Z/nZ$ are models of this theory. Now consider RT + $\forall x(Sx=x+1)$ + ...
2
votes
4answers
175 views

Why the priciple of strong induction stated in such way?

In most books,the principle of strong induction is stated as follows: Let $X$ be a well ordered set with an ordering relation $\leq$,and let $P(n)$ be a property pertaining to an element $n\in X$.Let ...
-1
votes
5answers
402 views

When to use transfinite induction?

How do we know when we are allowed to use transfinite induction in a proof ? Edit : considering the replies i should say the following Consider an infinite sum of fractions. By induction we can ...
2
votes
2answers
510 views

set-theoretic function definition; recursion theorem

I am an undergraduate student, currently studying axiomatic set theory (I am reading Halmos' Naive Set Theory as an overview, and consulting other sources recommended to me to supplement the sparser ...
6
votes
2answers
280 views

Does the principle of mathematical induction extend to higher cardinalities?

Does the principle of mathematical induction extend to a cardinality larger than that of the countably infinite?
1
vote
2answers
216 views

Strict ordering on natural numbers

I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way: They prove ...