- #1

MathematicalPhysicist

Gold Member

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now here a few a questions:

1. why cant you count in real arithmetic, surely you can count numbers (-: ?

2. what is this "model"?

3. why the "model" is countable but the arithmetic isnt?

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- Thread starter MathematicalPhysicist
- Start date

- #1

MathematicalPhysicist

Gold Member

- 4,443

- 265

now here a few a questions:

1. why cant you count in real arithmetic, surely you can count numbers (-: ?

2. what is this "model"?

3. why the "model" is countable but the arithmetic isnt?

- #2

Hurkyl

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Recall that "countable" when applied to a set means that the set can be placed into a 1-1 correspondence with the natural numbers. For example, with the rational numbers, we can write the enumeration

1 - 1/1

2 - 2/1

3 - 1/2

4 - 3/1

5 - 1/3 (we already have 2/2)

6 - 4/1

7 - 3/2

8 - 2/3

9 - 1/4

10 - 5/1

11 - 1/5

...

Every rational number will appear in this sequence, so the rational numbers are countable.

However, the real numbers are uncountable; it is impossible to make such an enumeration (via one of Cantor's diagonal arguments).

I presume by saying "real arithmetic is uncountable" it means that there are uncountably many real numbers.

As for the countable model... I presume that they mean when the axioms are weakened to be written in first-order logic. I don't know what one does to the axiom of completeness, but I know there's an important type of field called a

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