<s>
A	O
counting	B-Algorithm
quantifier	I-Algorithm
is	O
a	O
mathematical	O
term	O
for	O
a	O
quantifier	B-Language
of	O
the	O
form	O
"	O
there	O
exists	O
at	O
least	O
k	O
elements	O
that	O
satisfy	O
property	O
X	O
"	O
.	O
</s>
<s>
In	O
first-order	O
logic	O
with	O
equality	O
,	O
counting	B-Algorithm
quantifiers	I-Algorithm
can	O
be	O
defined	O
in	O
terms	O
of	O
ordinary	O
quantifiers	B-Language
,	O
so	O
in	O
this	O
context	O
they	O
are	O
a	O
notational	O
shorthand	O
.	O
</s>
<s>
Also	O
,	O
generalized	O
counting	B-Algorithm
quantifiers	I-Algorithm
that	O
say	O
"	O
there	O
exists	O
infinitely	O
many	O
"	O
are	O
not	O
expressible	O
using	O
a	O
finite	O
number	O
of	O
formulas	O
in	O
first-order	O
logic	O
.	O
</s>
<s>
Counting	B-Algorithm
quantifiers	I-Algorithm
can	O
be	O
defined	O
recursively	O
in	O
terms	O
of	O
ordinary	O
quantifiers	B-Language
.	O
</s>
