## New answers tagged ordinal-numbers

6
votes

Accepted

### Does negative trichotomy hold for constructive ordinals?

I'll assume by ordinal you mean a set with a transitive, well-founded, and extensional relation. Then it's not constructively valid that $\neg \neg (A < B \vee A = B \vee A > B)$ for ordinals $A$...

2
votes

### Does negative trichotomy hold for constructive ordinals?

It's three decades since I wrote Intuitionistic Sets and Ordinals, at which time I knew how to express well founded relations as coalgebras, but I had very little understanding of the subject.
In ...

6
votes

Accepted

### In the constructive theory of direct categories, is it decidable whether an arbitrary morphism is an identity or not?

I believe (2) is the best definition. Consider what you want to do with a direct category: you want to construct presheaves and natural transformations between them inductively, assuming that they ...

4
votes

### In the constructive theory of direct categories, is it decidable whether an arbitrary morphism is an identity or not?

To ask whether it is decidable if a morphism is an identity is the wrong question here. In any more general setting the answer to this ought to be "no", however, here we don't have any non-...

0
votes

### Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy

Notice that if we define operations on sequences element-wise as you suggest, and consider the regularized value of a sequence $a=(a_0,a_1,a_2,...)$ the following way:
$\operatorname{reg} a=\...

7
votes

### Embedding large countable ordinals into the complex plane

Here is another construction derived from a solution a student submitted to a final exam for my undergraduate set theory course some years ago.
Let $f:\alpha \hookrightarrow \mathbb{N}$ be an ...

6
votes

Accepted

### Embedding large countable ordinals into the complex plane

Yes, in fact every countable ordinal embeds into the rational numbers in this way, an order-preserving map as a closed set of rational numbers.
Let me give three proofs.
First, one can easily prove ...

6
votes

### End-extension which Mostowski collapses a fake well ordering

In Chapter 3 of his PhD thesis, Harvey Friedman showed that there is a recursive linear ordering $\prec$ which has no hyperarithmetic infinite descending sequence and which does not support a jump ...

2
votes

### Ordinals in constructive mathematics ? (references)

The first response to any question about "the" ordinals is what you want to do with them. Regrettably, that is most often to find the fixed point of some construction within some already-...

3
votes

### Ordinals in constructive mathematics ? (references)

If you want proof by induction, then you do want to use well-ordered sets, but you need the correct definition of ‘well-ordered’. Of course we can't require that any non-empty subset have a minimal ...

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