Formal developments in Geometry

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In summary, Alfred Tarski developed a first-order axiomatization for plane Euclidean geometry in his book INTRODUCTION TO LOGIC. This theory is decidable and does not require any set theory. It is based on the idea that two distinct points always determine a straight line. Tarski's axioms are more concise than Hilbert's and have only two primitive relations. However, there is no first-order development of geometry available on the internet.
  • #1
solakis1
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I wonder if we can have a 1st order Goemetry
 
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  • #2
solakis said:
I wonder if we can have a 1st order Goemetry
The logician Alfred Tarski devised a 1st order axiomatization for plane Euclidean geometry.
 
  • #3
caffeinemachine said:
The logician Alfred Tarski devised a 1st order axiomatization for plane Euclidean geometry.

Where in which book.
 
  • #4
solakis said:
Where in which book.
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.
 
  • #5
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.

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
 
  • #6
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

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
 
  • #7
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
According to wiki ,how then would we formalize the very 1st axiom of Geometry.

There is exactly one straight line on two distinct points
 
  • #8
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

Here's an axiomatization by Hilbert on geometry:
http://www.gutenberg.org/files/17384/17384-pdf.pdf

The actual axiom (apart from the necessary definitions) is:
I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a.​

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).
 

FAQ: Formal developments in Geometry

What is formal development in geometry?

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.

How is formal development different from classical geometry?

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.

What are the benefits of using formal development in geometry?

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.

Are there any limitations to formal development in geometry?

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.

How is formal development in geometry relevant to real-world applications?

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.

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