My Beautiful Proof Lies All In Ruins.

The Transfinite Numbers Are In A Certain Sense Themselves New Irrationalities And In Fact In My Opinion The Best Method Of Defining The Finite Irrational Numbers Is Wholly Disimilar To, And I Might Even Say In Priciple The Same As, My Method Described Above Of Introducing Trasfinite Numbers. One Can Say Unconditionally: The Transfinite Numbers Stand Or Fall With The Finite Irrational Numbers; They Are Like Each Other In Their Innermost Being; For The Former Like The Latter Are Definite Delimited Forms Or Modifications Of The Actual Infinite.

I Am So In Favor Of The Actual Infinite That Instead Of Admitting That Nature Abhors It, As Is Commonly Said, I Hold That Nature Makes Frequent Use Of It Everywhere, In Order To Show More Effectively The Perfections Of Its Author. Thus I Believe That There Is No Part Of Matter Which Is Not - I Do Not Say Divisible - But Actually Divisible; And Consequently The Least Particle Ought To Be Considered As A World Full Of An Infinity Of Different Creatures.

My Beautiful Proof Lies All In Ruins.

This View [of The Infinite], Which I Consider To Be The Sole Correct One, Is Held By Only A Few. While Possibly I Am The Very First In History To Take This Position So Explicitly, With All Of Its Logical Consequences, I Know For Sure That I Shall Not Be The Last!

I Realize That In This Undertaking I Place Myself In A Certain Opposition To Views Widely Held Concerning The Mathematical Infinite And To Opinions Frequently Defended On The Nature Of Numbers.