A Set Is A Many That Allows

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.

There Is No Doubt That We Cannot Do Without Variable Quantities In The Sense Of The Potential Infinite. But From This Very Fact The Necessity Of The Actual Infinite Can Be Demonstrated.

The Old And Oft-repeated Proposition "totum Est Majus Sua Parte" [the Whole Is Larger Than The Part] May Be Applied Without Proof Only In The Case Of Entities That Are Based Upon Whole And Part; Then And Only Then Is It An Undeniable Consequence Of The Concepts "totum" And "pars". Unfortunately, However, This "axiom" Is Used Innumerably Often Without Any Basis And In Neglect Of The Necessary Distinction Between "reality" And "quantity", On The One Hand, And "number" And "set", On The Other, Precisely In The Sense In Which It Is Generally False.

My Theory Stands As Firm As A Rock; Every Arrow Directed Against It Will Return Quickly To Its Archer. How Do I Know This? Because I Have Studied It From All Sides For Many Years; Because I Have Examined All Objections Which Have Ever Been Made Against The Infinite Numbers; And Above All Because I Have Followed Its Roots, So To Speak, To The First Infallible Cause Of All Created Things.

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