Plato, Meet Russell

Plato, Meet Russell

 

Plato spoke of ideal Forms, of which the entities around us are imperfect examples. Thus there are frogs, and there is the Form of all frogs, and the Form of all frogs is not a frog.

But there is also the Form of all Forms, which is itself a Form. Therefore some Forms are examples of themselves, and some are not.

Now consider Russell’s Form R, of all forms, and only those forms, that are not examples of themselves:

 

For any form F,  

F is an example of R    =    F is not an example of F

 

Is Russell’s Form R an example of itself? Substitution yields:

 

R is an example of R    =    R is not an example of R

 

A Russellian paradox!

Here is another paradoxical Form: G, the Groucho Form, which is the Form of all Forms, and only those Forms, that the Groucho form is not an example of:

 

For any form F,  

F is an example of G    =   G is not an example of F

 

Therefore:

 

G is an example of G    =    G is not an example of G

 

So the world of Platonic Forms is paradoxical or incomplete.