[FOM] The Field with One Element?

[Mario Carneiro]

On Fri, Jun 17, 2016 at 10:29 PM, Harvey Friedman <hmflogic at gmail.com>
wrote:

> DEFINITION. A pseudo field is a nonempty set F together with elements
> 0,1 (possibly the same) and binary functions +,dot, such that the
> following holds.
> 1. x+0 = x.
> 2. x+y = y+x.
> 3. x+(y+z) = (x+y)+z.
> 4. For all x, there exists y such that x+y = 0.
> 5. x dot 1 = x.
> 6. x dot y = y dot x.
> 7. x dot (y dot z) = (x dot y) dot z.
> 8. For all x not 0, there exists y such that x dot y = 1.
> 9. x dot (y+z) = x dot y + x dot z.
> 10. If x dot y = 0 then x = 0 or y = 0.
>

I think rule 10 is redundant. Since x1 + 0 = x1 = x(1+0) = x1 + x0 and
rules 1,3,4 give that F is a group and hence left-cancellative, x0 = 0.
Then suppose xy = 0 and y != 0. Let z be such that yz = 1, then x = x1 =
xyz = 0z = 0.

Mario
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