The Essence Of Mathematics Lies Precisely In

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.

I Entertain No Doubts As To The Truths Of The Tranfinites, Which I Recognized With God's Help And Which, In Their Diversity, I Have Studied For More Than Twenty Years; Every Year, And Almost Every Day Brings Me Further In This Science.

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.

What I Assert And Believe To Have Demonstrated In This And Earlier Works Is That Following The Finite There Is A Transfinite (which One Could Also Call The Supra-finite), That Is An Unbounded Ascending Lader Of Definite Modes, Which By Their Nature Are Not Finite But Infinite, But Which Just Like The Finite Can Be Determined By Well-defined And Distinguishable Numbers.

Mathematics Is Entirely Free In Its Development, And Its Concepts Are Only Linked By The Necessity Of Being Consistent, And Are Co-ordinated With Concepts Introduced Previously By Means Of Precise Definitions.