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.

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.

The Actual Infinite Arises In Three Contexts: First When It Is Realized In The Most Complete Form, In A Fully Independent Otherworldly Being, In Deo, Where I Call It The Absolute Infinite Or Simply Absolute; Second When It Occurs In The Contingent, Created World; Third When The Mind Grasps It In Abstracto As A Mathematical Magnitude, Number Or Order Type.

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The Transfinite Numbers Are In A Sense The New Irrationalities [ ... They] Stand Or Fall With The Finite Irrational Numbers.