Mathematics Is Entirely Free In Its Development,

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.

Mathematics, In The Development Of Its Ideas, Has Only To Take Account Of The Immanent Reality Of Its Concepts And Has Absolutely No Obligation To Examine Their Transient Reality.

To Ask The Right Question Is Harder Than To Answer It.

The Potential Infinite Means Nothing Other Than An Undetermined, Variable Quantity, Always Remaining Finite, Which Has To Assume Values That Either Become Smaller Than Any Finite Limit No Matter How Small, Or Greater Than Any Finite Limit No Matter How Great.

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.