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tgt
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For me it would have to be the set of complex numbers. What's yours?
mathman said:Mathematics starts with the integers, so I would consider them the most important.
turbo-1 said:The concept of zero, hands-down.
mathman said:Mathematics starts with the integers, so I would consider them the most important.
tgt said:For me it would have to be the set of complex numbers. What's yours?
Office_Shredder said:From the empty set you can construct everything.
n!kofeyn said:I thought the empty set is just a convention, unless this was a joke, but I've seen seconds and thirds of the empty set.
Office_Shredder said:The empty set is the set containing no elements. How is that a convention? Using the standard set theory axioms, given the empty set you can 'construct' the natural numbers, and hence basically all of math
n!kofeyn said:Well this is how James Munkres describes it in his Topology book. I have also seen other books take it as convention. I'm not familiar with foundational set theory, but from what I've read, the empty set is just assumed to exist, either by definition, convention, or axiom. I realize that it is defined as the set containing no elements, but how do you know such an object even exists?
Office_Shredder said:In foundational set theory it's assumed to exist as an axiom. How do you know any object exists? Nothing exists in math without assuming something, so I wouldn't call it a weakness
n!kofeyn said:I didn't say it is a weakness. Many objects in math are shown to actually exist, although everything is based upon the basic assumed axioms. I think my point is that what I seem to understand is that the empty set is just assumed to exist by axiom because it makes things easier to talk about. In other words, it isn't absolutely necessary, but it is a convention that simplifies the discussion of some things.
Pinu7 said:Logical Quantifiers.
Mathematicians are too lazy to type out "for every" and would rather go to a mathtype window to make an upside-down "A" so their theorems are unreadable to those that do not know logical quantifiers.
maverick_starstrider said:*insert offensive generalization about hygiene, women, etc.*.. oh snap!
Ok, but then nothing is necessary because you can just change the axioms to make sure they don't exist. So I'm still unsure how the empty set is special in this regard.
The single most important object in mathematics is the number 1. This number is the basis for all other numbers and operations in mathematics.
The number 1 is considered the most important object in mathematics because it is the identity element for multiplication and division, and the neutral element for addition and subtraction. It is also the basis for counting and the foundation for all other numbers.
The number 1 is essential for understanding concepts such as fractions, decimals, and percentages. It is also used in mathematical operations like exponents and logarithms.
The number 1 is used in many real-world applications, such as measuring quantities, calculating ratios and proportions, and representing probabilities. It is also used in financial calculations, such as interest rates and inflation rates.
No, the number 1 is not the only important object in mathematics. Other important objects include zero, pi, and infinity, among others. However, the number 1 holds a special significance as the most fundamental and essential object in mathematics.