Collection of Formulas of Set Theory (Symbols)

In summary, the symbols φx, ψx, φ(x,y), etc. denote propositional functions, while the symbols φxˆ, ψxˆ, φ(xˆ,zˆ) denote terms for propositional functions.
  • #1
heff001
30
1
In 'AN INTRODUCTION TO SET THEORY' by Professor William A. R. Weiss, the following symbols are applied to the snippet below:
...
The collection of formulas of set theory is defined as follows:
1. An atomic formula is a formula.
2. If ϕ and Ψ are formulas, then (ϕ ^ Ψ) is also a formula.
...

I am new to Set Theory and my 'greenhorn' question really is about the use of these specific symbols here (I understand the Basics). In his book, has he assigned the Greek symbols shown above and states we have now specified a 'language' of set theory. Can someone please explain what the bolded Capital Phi ϕ symbol and Ψ symbol represent specifically in the field of Set Theory, or has he just applied them in his book purely as formula symbols as 'his' specification of set theory 'language'.

Never mind - see
http://www.mathacademy.com/pr/prime/articles/greek/index.asp

The Greek Alphabet

Mathematics requires a large number of symbols to stand for abstract objects, such as numbers, sets, functions, and spaces, so the use of Greek letters was introduced long ago to provide a collection of useful symbols to supplement the usual Roman letters.
To us these symbols may seem quite foreign, and they are difficult to become familiar with. However, at the time they were introduced, most scholars had been taught at least some Latin and Greek during their education, so the letters did not seem nearly so strange to them as they do to us. Since then, each new generation of mathematicians has just gotten used to using them.
The Greek alphabet contains lower-case letters that are used more often than the upper-case letters, but the latter are used often enough. The lower-case letters are most often used for variables, such as angles and complex numbers, and for functions and formulas, while the upper-case letters more commonly stand for sets and spaces, and sometimes for repeated arithmetic operations such as adding and multiplying (see Sigma and Pi). In any particular textbook or paper, the way in which these symbols should be interpreted should generally be clear from the context and definitions.
 
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  • #2
Generally, ϕ is used to denote a property. Eg. ϕ(x) is either True or False. I'm learning.
 
  • #3
heff001 said:
Can someone please explain what the bolded Capital Phi ϕ symbol and Ψ symbol represent specifically in the field of Set Theory
Whatever usual meanings they might have had are overridden when he specifically states that, in that sentence, they are are variable symbols that denote arbitrary formulas.
 
  • #4
Where can I find more about the basics of set theory formulas (specifically ϕ symbol and Ψ symbol definitions and usage)? Is it best to start in Logic or Mathematical Foundations books? Please send me any references.
 
  • #5
heff001 said:
Where can I find more about the basics of set theory formulas (specifically ϕ symbol and Ψ symbol definitions and usage)? Is it best to start in Logic or Mathematical Foundations books? Please send me any references.

There is no special meaning to ϕ and Ψ: it means exactly what Weiss said it means. The symbols are irrelevant: it could have just as well been

The collection of formulas of set theory is defined as follows:
1. An atomic formula is a formula.
2. If FISH and CAT are formulas, then (FISH ^ CAT) is also a formula.

or

The collection of formulas of set theory is defined as follows:
1. An atomic formula is a formula.
2. If
20px-Prince_symbol.svg.png
and ƒ are formulas, then (
20px-Prince_symbol.svg.png
^ ƒ) is also a formula.
 
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  • #6
I thank you for your simple but stark reply, but originating accepted standard notations and terminology are adhered to / used in many textbooks.

I discovered, in the Principia Mathematica [PM] system of symbolic logic, the definition of mathematical notions in terms of logical notions, used to prove the fundamental axioms of mathematics as theorems of logic. Unfortunately, over time, this notation has become alien to contemporary students of logic, which has become a barrier to the study of Principia Mathematica. The Stanford University site provides a partial translation of the PM symbolism into a more contemporary notation which they state is quite standard in contemporary textbooks of symbolic logic. I listed a snippet below specifically around my initial new-bee type question.

Primitive Symbols

φ, ψ, χ, etc., and f, g, etc. are variables which range over propositional functions, no matter whether those functions are simple or complex.

φx, ψx, φ(x,y), etc. are open atomic formulas in which both ‘x’ and ‘φ’ are free. [An alternative interpretation is to view ‘φx’ as a schematic letter standing for a formula in which the variable ‘x’ is free.]

φxˆ, ψxˆ, φ(xˆ,zˆ), etc. are terms for propositional functions. Here are examples of such terms which are constants: ‘xˆ is happy’, ‘xˆ is bald and xˆ is happy’, ‘4 < xˆ < 6’, etc. If we apply, for example, the function xˆ is bald and xˆ is happy to the particular individual b, the result is the proposition b is bald and b is happy.

I am moving on...I've learned my lesson here...
 

Related to Collection of Formulas of Set Theory (Symbols)

What is Set Theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects. It provides a formal language for describing and analyzing relationships between sets.

What are the symbols used in Set Theory?

The symbols used in Set Theory include the following:
• ∈ - denotes membership
• ∉ - denotes non-membership
• ⊆ - denotes subset
• ⊈ - denotes not a subset
• ∪ - denotes union
• ∩ - denotes intersection
• ∖ - denotes set difference
• ∅ - denotes empty set
• ℕ - denotes natural numbers
• ℤ - denotes integers
• ℚ - denotes rational numbers
• ℝ - denotes real numbers
• 𝔸 - denotes arbitrary set
• ⊂ - denotes proper subset
• ⊄ - denotes not a proper subset
• ⊊ - denotes not a proper subset or equal to
• ⊃ - denotes superset
• ⊅ - denotes not a superset
• ⊋ - denotes not a superset or equal to

How are sets represented in Set Theory?

Sets are typically represented using curly braces, {} , and the elements inside are separated by commas. For example, the set of all even numbers can be represented as {2, 4, 6, 8, ...}.

What is the difference between a set and a subset?

A set is a collection of distinct objects, while a subset is a set that contains elements that are all part of a larger set. In other words, all elements in a subset are also elements of the larger set, but not all elements of a set are necessarily elements of a subset.

How is set equality determined in Set Theory?

Two sets are considered equal if they have the same elements. This means that if two sets have the same elements, regardless of order or repetition, they are considered equal. For example, the sets {1, 2, 3} and {3, 2, 1} are equal sets.

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