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- Thread starter solakis
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- Mar 10, 2012

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The logician Alfred Tarski devised a 1st order axiomatization for plane Euclidean geometry.I wonder if we can have a 1st order Goemetry

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Where in which book.The logician Alfred Tarski devised a 1st order axiomatization for plane Euclidean geometry.

- Mar 10, 2012

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I don't know that. I just know that he did. If you want to read the theory then you have to google about a bit.Where in which book.

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Alfred Tarski in his book: INTRODUCTION TO LOGIC he developed the axiomatic theory for real Nos and nothing about Geometry.I don't know that. I just know that he did. If you want to read the theory then you have to google about a bit.

No where in the internet there is a first order development of Geometry

- Mar 10, 2012

- 835

I quote from wiki,Alfred Tarski in his book: INTRODUCTION TO LOGIC he developed the axiomatic theory for real Nos and nothing about Geometry.

No where in the internet there is a first order development of Geometry

"In the 1920s and 30s, Tarski often taught high school geometry. Using some ideas of Mario Pieri, in 1926 Tarski devised an original axiomatization for plane Euclidean geometry, one considerably more concise than Hilbert's. Tarski's axioms form a first-order theory devoid of set theory, whose individuals are points, and having only two primitive relations. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers."

See Alfred Tarski - Wikipedia, the free encyclopedia

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According to wiki ,how then would we formalize the very 1st axiom of Geometry.I quote from wiki,

"In the 1920s and 30s, Tarski often taught high school geometry. Using some ideas of Mario Pieri, in 1926 Tarski devised an original axiomatization for plane Euclidean geometry, one considerably more concise than Hilbert's. Tarski's axioms form a first-order theory devoid of set theory, whose individuals are points, and having only two primitive relations. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers."

See Alfred Tarski - Wikipedia, the free encyclopedia

There is exactly one straight line on two distinct points

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- Mar 5, 2012

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Here's an axiomatization by Hilbert on geometry:According to wiki ,how then would we formalize the very 1st axiom of Geometry.

There is exactly one straight line on two distinct points

http://www.gutenberg.org/files/17384/17384-pdf.pdf

The actual axiom (apart from the necessary definitions) is:

Note that points and lines are just abstract concepts that are referenced by the axioms.

They don't have to be anything like real-life points or lines.

For instance, a line might actually be a plane (in projective geometry).