# Questions tagged [transfinite-induction]

The tag has no usage guidance.

54 questions
31 views

62 views

### Logical question while trying to prove a theorem about comparing well-ordered sets

I'm reading Set theory of Jech. I try to prove Thm 2.8 for myself but I'm not sure if the first step of my proof is correct. So the theorem affirms that if $W_1$ and $W_2$ are two well-ordered sets, ...
226 views

### Transfinite induction (Atiyah and Macdonald)

Exercise 17 in chapter 4 of Introduction to Commutative Algebra by Atiyah and Macdonald hints for you to use transfinite induction to complete the proof. I have not come across transfinite induction ...
110 views

### Hahn Banach Theorem: Transfinite Induction

In the 4th book of Stein and Shakarchi, the statement of the Hahn-Banach theorem is as follows: Suppose $V_0$ is a linear subspace of $V$ and $p$ is a real sub-linear function on $V$, and that we ...
94 views

### Union of Connected Sets and Transfinite Induction

Recall the following theorem: Theorem Let $\{A_\alpha\}$ be a collection of connected subsets of $X$ such that $\bigcap_\alpha A_\alpha\neq\emptyset$. Then $\bigcup_\alpha A_\alpha$ is a ...
527 views

### Closed kernel implies continuous linear functional : Zorn's Lemma

I know that this question has been asked before, but I am thinking of a solution using Zorn's lemma. Stein and Shakarchi's volume 4 wants me to prove the above statement(exercise 35) using exercise 34:...
67 views

### Transfinite induction on inductively constructed CW complex

Let $X$ be a 2-dimensional path-connected CW-complex and let $W \subset X$ be a subcomplex. Further let $I$ be an index set of the path-components of $W$. Consider the following iterative procedure: ...
142 views

62 views

### Transfinite Moduli

I'm looking at arithmetic $\mod ω$ (where $ω := |**N**| = 1+1+1+... =$ Lim($n$) : n ∈ N). Specifically, I'm trying to show that $... + \frac 18 + \frac 14 + \frac 12 + 1 + 2 + 4 + 8 + ... ≡ 0 \mod ω$...
53 views

### Is this a feasible way to show that a group $G$ is isomorphic to its opposite group $G^{op}$?

Let $G$ be a group with group operation $*$. Let $G^{op}$ be a group that shares the same base set with $G$ but the group operation $*^{'}$ is such that $a*^{'}b = b*a$. We would like to show that $G$ ...
76 views

### Induction, coinduction, and ordinal induction

As a computer scientists, I have been thought to use induction and coinduction principles for proving properties of finite and infinite structures, respectively. When does one use ordinal induction?...
287 views

### Using Strong Induction

The statement is: If $n$ is an integer with $n > 1$, then $n$ has a prime factor. I know you have to use $P(k+1)$ but I'm not sure how to implement that into the proof. Is this even going in the ...
199 views

41 views

### Could you verify my proof by transfinite induction of: $\langle a_\xi : \xi< \alpha\rangle =\langle b_\xi : \xi< \alpha\rangle$?

I'm am trying to comprehend the proof on page 22 of Thomas Jech's Set theory and work out my first ever proof by transfinite induction. The problem is this: Le $G$ be a function (on $V$), then ...
75 views

### Using transfinite induction to compute fundamental groups

I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen. Let's ...
580 views

### How does one prove transfinite induction in ZFC?

If one is able to use classes, it seems to me that the proof of transfinite induction is a simple extension of the usual proof of induction (and equal to the proof of transfinite induction on sets). ...
51 views

### “Hahn decomposition” for countable or arbitrarily many mutually singular probability measure

Let $(\mu_i)_{i\in I}$ be a collection of mutually singular probability measure on $(\Omega, \mathcal F)$.(i.e. any two of them are singular to each other). I am not sure if the following statement is ...
276 views

### Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
2k views

### What are the most prominent uses of transfinite induction outside of set theory?

What are the most prominent uses of transfinite induction in fields of mathematics other than set theory? (Was it used in Cantor's investigations of trigonometric series?)
### Theorem $($Transfinite Induction and Construction$)$ (James Dugundji - 5.1, Page - 40)
Let $W$ be a well ordered set, and let $Q \subset W$. If $[ W(x) \subset Q] \Rightarrow [ x \in Q]$ for each $x \in W$, then $Q = W$. For each $a \in W$, the set \$W(a) = \{x \in W; (x\prec a) \...