"THERE EXISTS" PHRASES

Variables in mathematics, such as x, y, a, b, c, etc., are quantified with phrases such as "for all x", "for any z", "for every z", "there is at least one a", "there is some b", "there is an x", "there is a w", "for at least one j", etc. "There is at least one", "there is some", "there is an", "there is a", "for at least one", and similar phrases mean the same thing: there is at least one object with the property being described and MAYBE more. No claim is being made about there being more than one such object: there might be exactly one, there might be more than one.

• There is a real number x such that, for all real numbers y, x+y=y. This sentence mixes two kinds of quantifying ("there is" and "for all"). When the "there is" comes first, it means that there is at least one such object that has some property for ALL of the second variable under discussion. In this example, x=0 has the property that, for all real numbers y, 0+y=y.
• There is some real number x such that, for all real numbers y, x*y=y. Note that x=1 has the desired property that, for all real numbers y, 1*y=y.
• For all real numbers x, there is a real number y such that x+y=0. This sentence also mixes two kinds of quantification ("for all" and "there is"). With the "for all" coming first, the y that is required to exist IS ALLOWED TO BE DIFFERENT FOR EACH x. So, this statement is true because y=-x has the property that x+y=0.
• For all real numbers x, there is a real number y such that x*y=1. This sentence is false, because it happens to have just one exception: when x=0, x*y=0 for all real numbers y and there is no way to get some y so that 0*y=1.
• For all non-zero real numbers x, there is a real number y such that x*y=1. This sentence is true, because for non-zero x we can let y=1/x. Note that x*(1/x)=1.
• For all positive real numbers x, there is some real number y such that y*y=x. In this example, there are in fact two such y's: the square root of x and the negative of the square root of x.
• For all real numbers x, there is some real number y such that y*y*y=x. In this example, there is exactly one such y---namely, the real number cube root of x.
• For all E>0, there is some D>0, such that if |x-c| < D then |f(x)-f(c)| < E. This sentence is equivalent to the continuity of f at c.

Consider these two sentences: "There is an x and there is a y such that x+y=y" and "There is a y and there is an x such that x+y=y". These mean the same thing. Order is not important with multiple "there is" phrases. This is NOT true for mixtures of "there is" and "for all" phrases.

Compare these two sentences: "For all x, there exists at least one y such that Q(x,y)" and "there is at least one y such that, for all x, Q(x,y)". Here Q(x,y) is some sentence about x and y, such as "x+y=0". The first of these means that for all x, there is a y WHICH IS ALLOWED TO VARY WITH EACH x which makes Q(x,y) true. So, for example, "for all x, there exists at least one y such that x+y=0" is true because y=-x makes it true.

The sentence "there is at least one y such that, for all x, Q(x,y)" means that there exists one y which does the job for all x of making Q(x,y) true. This is a much tougher statement to make true! Notice that "there is at least one y such that, for all x, x+y=0" is false for the real numbers---there y has to be -x and must vary with x which this sentence does not allow.

Of course the tougher statement implies the easier. Consider "there is a z such that, for all x, x+z=x". This is true for real numbers, because z=0 plays that role for ALL x. It is likewise true that, "for all x, there is at least one z such that x+z=x." While z is allowed to vary with x to make this last sentence true, it does not have to vary---z=0 still works for each and every x! To summarize this paragraph, the following is always true:

If [there is at least one y such that, for all x, Q(x,y)], then [for all x, there is at least one y such that Q(x,y)].