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solakis1
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I wonder if we can have a 1st order Goemetry
The logician Alfred Tarski devised a 1st order axiomatization for plane Euclidean geometry.solakis said:I wonder if we can have a 1st order Goemetry
caffeinemachine said:The logician Alfred Tarski devised a 1st order axiomatization for plane Euclidean 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.solakis said:Where in which book.
caffeinemachine said: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.
solakis said: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
According to wiki ,how then would we formalize the very 1st axiom of Geometry.caffeinemachine said: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
solakis said:According to wiki ,how then would we formalize the very 1st axiom of Geometry.
There is exactly one straight line on two distinct points
Formal development in geometry is a systematic approach to the study of geometric concepts and properties using logical reasoning and mathematical proofs. It involves defining axioms and postulates, and then using deductive reasoning to derive new theorems.
Formal development in geometry differs from classical geometry in that it is based on a rigorous and axiomatic approach rather than intuitive reasoning. It also utilizes symbolic notation and logical reasoning to prove theorems, rather than relying on visual aids.
Using formal development in geometry allows for a deeper understanding of geometric concepts and properties, as well as the ability to prove new theorems using logical reasoning. It also provides a solid foundation for further studies in mathematics and other scientific fields.
One limitation of formal development in geometry is that it can be time-consuming and complex, requiring strong mathematical skills. It may also be difficult to visualize geometric concepts without the aid of diagrams and visual aids.
Formal development in geometry has many practical applications, such as in architecture, engineering, and computer graphics. It also provides a logical framework for understanding and solving real-world problems involving geometric concepts and properties.