# Mathematical meaning of 'as many as'

1. Jan 27, 2016

### DaveC426913

Is this a valid statement?

There are a hundred times as many donuts in box A as in box B.

A B

i.e. 0x100=0

I mean, I know it's a disingenuous statement, but is it valid?

2. Jan 27, 2016

### Staff: Mentor

It is as any statement about the elements of the empty set is true. I like to say: the elements of the empty sets have purple eyes.

3. Jan 27, 2016

### DaveC426913

So, one could say that there are a hundred times as many known pink unicorns in the galaxy as there are known alien megastructures.

4. Jan 27, 2016

### Staff: Mentor

Every existential statement about the elements of the empty set is false, e.g. there is an $x ∈ \{\}$ such that ..., for there is no element to fulfill the condition.
Every for-all statement about the elements of the empty set is true, e.g. for all $x ∈ \{\} : x ...$, for there is no element for which the condition has to be proven.
However, as soon as you talk about the cardinality of sets you start to talk about arithmetic and not about the elements of the empty set anymore. The cardinal of the empty set is clearly $0$. So $0 = |\{known-pink-unicorns\}| = 100 \cdot |\{known-alien-megastructures\}| = 100 \cdot 0$.

5. Jan 27, 2016

### DaveC426913

Er... so, yes?

6. Jan 27, 2016

### Staff: Mentor

Yes.

7. Jan 27, 2016

### Staff: Mentor

Yes, valid. There are (apparently) zero donuts in box A, and 100 * 0 in box B.

I have to say, though, that it looks like there is only one box, so it would also be valid to say "the box" contains 100 times as many donuts as are contained in itself.

8. Jan 27, 2016

### DaveC426913

They're just Googled images for illustrative effect.

9. Jan 27, 2016

### DaveC426913

Cool. Thanks.

I'm tired of people engaging in the finer details of what kind of megastructure aliens might be building around Tabby's star to explain the dimming.

I've got my own theory, involving a cosmic pink unicorn, for which there is as much as 100 times the evidence as for alien megastructures.

10. Jan 27, 2016

### Staff: Mentor

C'mon, you must admit that funny star is a good mystery. And with the data being as they are at the current moment you as well could be right with your pink unicorn. However, I thought it is invisible.

11. Jan 31, 2016

### WWGD

I don't mean to be pedantic about this, but I think it may be helpful to point out that validity is usually an attribute of arguments. I think the question to be asked is whether the statement is true. And it is . Or may be you can frame this as an argument: But I think then you then need predicate logic to write this in.

12. Jan 31, 2016