There Is No Doubt That We Cannot

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.

The Essence Of Mathematics Lies Precisely In Its Freedom.

In Mathematics The Art Of Proposing A Question Must Be Held Of Higher Value Than Solving It.

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.

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