Mathematics Is Entirely Free In Its Development,

Every Transfinite Consistent Multiplicity, That Is, Every Transfinite Set, Must Have A Definite Aleph As Its Cardinal Number.

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 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.

The Fear Of Infinity Is A Form Of Myopia That Destroys The Possibility Of Seeing The Actual Infinite, Even Though It In Its Highest Form Has Created And Sustains Us, And In Its Secondary Transfinite Forms Occurs All Around Us And Even Inhabits Our Minds.

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!