of arithmetic. This seemed to require a truth definition for arithmetic, which in turn led to paradoxes, such as the Liar paradox (‚ÄúThis sentence is false‚Äù) and Berry‚Äôs paradox (‚ÄúThe least number not defined by an expression consisting of just fourteen English words‚Äù). G√∂del then noticed that such paradoxes would not necessarily arise if truth were replaced by provability. But this means that arithmetic truth', metadata={'Header 1': 'Kurt G√∂del', 'Header 2': '2. G√∂del‚Äôs Mathematical Work', 'Header 3': '2.2 The Incompleteness Theorems', 'Header 4': '2.2.1 The First Incompleteness Theorem'}),     Document(page_content='means that arithmetic truth and arithmetic provability are not co-extensive ‚Äî whence the First Incompleteness Theorem.', metadata={'Header 1': 'Kurt G√∂del', 'Header 2': '2. G√∂del‚Äôs Mathematical Work', 'Header 3': '2.2 The Incompleteness Theorems', 'Header 4': '2.2.1 The First Incompleteness Theorem'}),     Document(page_content='This account of G√∂del‚Äôs discovery was told to Hao Wang very much after the fact; but in G√∂del‚Äôs contemporary correspondence with Bernays and Zermelo, essentially the same description of his path to the theorems is given. (See G√∂del 2003a and G√∂del 2003b respectively.) From those accounts we see that the undefinability of truth in arithmetic, a result credited to Tarski, was likely obtained in some form by G√∂del by 1931. But he neither publicized nor published the result; the biases logicians', metadata={'Header 1': 'Kurt G√∂del', 'Header 2': '2. G√∂del‚Äôs Mathematical Work', 'Header 3': '2.2 The Incompleteness Theorems', 'Header 4': '2.2.1 The First Incompleteness Theorem'}),     Document(page_content='result; the biases logicians had expressed at the time concerning the notion of truth, biases which came vehemently to the fore when Tarski announced his results on the undefinability of truth in formal systems 1935, may have served as a deterrent to G√∂del‚Äôs publication of that theorem.',
Description and motivation
Description and motivation ->: of arithmetic. This seemed to require a truth definition for arithmetic, which in turn led to paradoxes, such as the Liar paradox (‚ÄúThis sentence is false‚Äù) and Berry‚Äôs paradox (‚ÄúThe least number not defined by an expression consisting of just fourteen English words‚Äù). G√∂del then noticed that such paradoxes would not necessarily arise if truth were replaced by provability. But this means that arithmetic truth', metadata={'Header 1': 'Kurt G√∂del', 'Header 2': '2. G√∂del‚Äôs Mathematical Work', 'Header 3': '2.2 The Incompleteness Theorems', 'Header 4': '2.2.1 The First Incompleteness Theorem'}),     Document(page_content='means that arithmetic truth and arithmetic provability are not co-extensive ‚Äî whence the First Incompleteness Theorem.', metadata={'Header 1': 'Kurt G√∂del', 'Header 2': '2. G√∂del‚Äôs Mathematical Work', 'Header 3': '2.2 The Incompleteness Theorems', 'Header 4': '2.2.1 The First Incompleteness Theorem'}),     Document(page_content='This account of G√∂del‚Äôs discovery was told to Hao Wang very much after the fact; but in G√∂del‚Äôs contemporary correspondence with Bernays and Zermelo, essentially the same description of his path to the theorems is given. (See G√∂del 2003a and G√∂del 2003b respectively.) From those accounts we see that the undefinability of truth in arithmetic, a result credited to Tarski, was likely obtained in some form by G√∂del by 1931. But he neither publicized nor published the result; the biases logicians', metadata={'Header 1': 'Kurt G√∂del', 'Header 2': '2. G√∂del‚Äôs Mathematical Work', 'Header 3': '2.2 The Incompleteness Theorems', 'Header 4': '2.2.1 The First Incompleteness Theorem'}),     Document(page_content='result; the biases logicians had expressed at the time concerning the notion of truth, biases which came vehemently to the fore when Tarski announced his results on the undefinability of truth in formal systems 1935, may have served as a deterrent to G√∂del‚Äôs publication of that theorem.',